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OUTLINES 


1S Pe 


NUMBER SCIENCE 


aati» 5 aes 


NATHAN NEWBY, 


Professor of Mathematics in the 


INDIANA STATE NORMAL SCHOOL, 


TERRE BAUTE, JNDIANA. 


0 ome Po oe — 


THIRD EDITION. 


pee gepordine to die of Congress in the year 


ea NATHAN NEWBY. Lien 


Al 


adade 2) Mu 


NA ee MATHEMATICS LIBRARY 


TABLE OF CONTENTS. 


Number Genesis.—Page 9. 
Number Classification.—16. 


Integers and Fractions 16 | Simple and Denomi- 
Abstract and Concrete 16 TEATOR SC Ptr nen 17 


Composite and Prime. 17 | Compound .......... 17 
Positive and Negative. 18 


Number Representation.—19. 


PPOUAAOM a's. se ek 19 | Systems of Numbers.. 21 
i Sa Anes 1gad Decimal ax a ass een 21 
ESO UAN . .  .). . - + oe 19° bbe Vractional se. 27 
GAT DIC: .*.:!.. 2A; es Compound i255 hy is 28 
Number Reduction.—29. 
Descending ......... 29'| Ascending .. 0.0)... 30 
Number Processes.—35. 
Synthetic. Analytic. 
PERAUOLEOD.. .)). <2). 36 | a. Subtraction ....... 56 
6b. Multiplication..... AGG Os ML VAIBIOIU ss 4 see 63 
c. Composition....... 621. cs Disposition | e224. (2 
PeANVOUIION . 2.2.25: 53 | d. Evolution ........ ris) 
Divisors and Multiples.—77. 
PT os ges ee e's Fa MEE GS Bas eet i 80 
Fractions—82. , 
Pumaary..1dea....... 3 82 | Addition and Subtrac- 
Classes of Fractions... 83 fractiows: waewe sie nde 
General Principles.... 85 | Multiplication........ 93 — 
BROALCTAOTIG | Sissi 3 (2s, SG FAP VISION ety eee 96 
PVUGMOUS SS 2 isan eras icias 105 


464214 


4 CONTENTS. 


Compound Numbers—Page 107. 


PRACTAM AG tad od dere pases 107 | Application to Time 
Order of Study: ...... 108 Measure. .... TOS 


Time and Longitude—119. 
Areas and Volumes—124. 
Metric System—128. 
Percentage—137 


First Class. Second Class. 
Profit. and’ Loss: §.achsk. 143, | Interest... eee 164 
CCOMDINISSTOM MG Ot ieee 149: +: Discountecnieouies AMS ops § 
SED OACS Coie eek So b44 ok: pea vee vg 
PVRAIPAN PO, ee eatias 159465 Bankice . eee rs 
PLEA ie 5 LWA tata 161 | Exchange...... .176 
Customs j.5 904.7 ... 163 | Equation of Payments. 177 
| Ratio and Proportion—180. 
Pre LO MGA tale k ale Sime 180 |. Proportion. .\. .. 22a 182 


Involution and Evolution—190. 


Evolution of Third 
Root. 2 2.00 


Test Problems—202. 
Review Questions and Topics—214. 
Appendix—223. 


Evolution of Second 


PFRERACE, 


This book is prepared to meet the author’s conven- 
ience in giving arithmetical instruction in the Normal 
School, and at the solicitation of a large number of 
pupil-teachers who have received in the class-room and 
have afterwards used in their own schools, much of 
the matter herein contained. 

“The aim of the Normal School is not so much to 
teach the facts of the common school branches as to, 
make a thorough study of the relations of those facts 
to one another. The study of these relations opens up 
new lines of thought that make the common school 
branches intensely interesting studies to most stu- 
dents.” 

The province of the school as thus stated, being kept 
in view, the attempt has been made so to present the 
topics of number science that the lines which unify 
parts that are kindred, may be readily seen. 

An examination of the order of procedure will show 
that the book is not made for children, but for the 
mature mind already conversant with most of the facts 
of number as presented in the text books on Arith- 
metic. , 

The discussion begins with a general definition of 
Mathematics, and an allusion to the realms of Space 


6 PREFACE. 


and Zime as being basal to the subject: Space as the 
condition of extension, and hence preliminary to 
Geometry in its various phases; and Time as the con- 
dition of succession, and hence preliminary to the idea 
of Number. 


Starting with Time as the conditioning factor, it is 
sought to show the genesis of number in general. 
From this idea the mind readily passes to that of num- | 
ber in particular. The term Integral unit or Unit one 
(from Davies), is fixed upon to designate the primary 
idea of number in particular. Itmustbe borne in mind 
that the elements which enter into a science are mental. 
and not material objects. A pencil, a horse or a box, is 
not an element of number science—is not a wnt; it is 
the idea one which the mind forms upon viewing the 
object as an entirety that constitutes the fundamental 
element in the science of numbers. 


The student is next asked to re-think the general 
classes of numbers, to find the basis of each classifica- 
tion, and to give, by definition, the mark of each class. 


The classification of numbers is discussed thus early 
not because of its logical relation to that which pre- 
cedes or succeeds it, but because of the basal character 
which Number classification sustains in all computa- 
tion. 


Number Representation is next discussed. Under 
the Arabic Notation it is observed that the characters 
are used to represent numerical values thought in 
three systems of numbers. Each of these systems, to- 
gether with its notation, is discussed. 


Number Reduction is next considered. The kinds 
of reduction are determined, and applied to numbers 


PREFACE. yey 


thought in the decimal, the fractional and the com- 
pound systems. 


Under Number Processes, the phases of synthesis 
and analysis are treated. The terms used, the mental 
acts involved, and the principles which guide the 
mind in computing are discussed. 


In formulating definitions and principles it is 
sought in the main, to “bring before the mind the act 
or process by which the concept to be defined is sup- 
posed to be constructed.” It has been a special aim to 
make every definition sufficiently inclusive to embrace 
all that the term covers wherever found in the work; 
e. g., the definitions of multiplication and division as 
given in most text books on Arithmetic are but par- 
tial since they do not include multiplication and divi- 
sion by a fraction. The definitions in this book are be- 
lieved to be ample. 


Both common and decimal fractions are treated 
together, the principles of the one being the principles 
of the other. 


The suggestions given for the treatment of Com- 
pound Numbers, will, it is hoped, enable the teacher 
to proceed more systematically and satisfactorily than 
by the methods usually presented. 


Most of the definitions and discussions’under the 
applications of Percentage are omitted, not because 
they are unimportant, but because they are so well 
given in text books on Arithmetic that their repeti- 
tion here is deemed unnecessary. 


Methods of solving representative problems have 
been freely inserted under these applications. 


8 PREFACE. 


Involution and Evolution are treated Arithmetic- 
ally instead of Geometrically and in a manner at once 
simple and exhaustive. 


“Results in teaching depend upon the clearness 
with which distinctions are made ;” and the pupil can 
be brought to make clear distinctions only by being 
held to rigidly logical modes of thinking. As an aid 
to this end, forms of solution are given for nearly all 
classes of arithmetical exercises. These forms or others 
equally logical should be strictly adhered to in order 
to secure to the pupil the maximum culture which 
the subject can give. ; 


A number of errors, typographical and otherwise, 
were observed after the work was in print. Some of 
these have been corrected with the pen. Others still 
remain, but they are of such a character as not to mis- 
lead the attentive reader. 


TERRE Haute, Inp., September, 1886. 


OUTLINES 


—-ON--— 


Nd&MBER SGIENGE. 


CHAPTER 1. 


[Notgs. 1. Mathematics may be defined in a general way 
as the department of knowledge which exhibits the properties 
and relations of extension and number. 

2 A consideration of extension leads into the realm of space. 

Geometry and kindred branches are evolved from the prop- 
erties and relations of extension. 

3. A consideration of number leads into the realm of time ag 
the conditioning factor. ] 


NUMBER GENESIS. 


1. In General. 1. Every conscious mental state is 
known to begin, to endure for a period and to end, giv- 
ing way to another which is distinct from that which 
preceded it. 

| a 


) fe) NUMBER GENESIS. 


2. Mental states are known to be distinct from 
‘one another because experienced in different periods 
of time or because of a diversity of the states compared. 


3. In transferring the attention from a given state 
to a state that is distinct from the given state, the mind 
becomes cognizant of succession. 


In the cognition of succession seems to arise the 
knowledge of number. 


Time is the condition of succession; hence time is 
primarily conditional for the idea of number. 


2. In Particular. If the attention be directed toan 
object, as a tree, a house, an apple, etc., among the 
attributes observed is that of oneness. The idea one 
may be abstracted from the conception of the object 
which occasions it, and as thus abstracted, it may be 
thought apart from its object. 

1. THE Primary Unir. The idea one as abstracted 
from the conception of an object thought as a whole, as 
not related to any other whole by being either a part of 
another whole or as made up of other wholes, is called 
the integral unit, or unit one. (From Davies.) 

2. Seconpary Units. a. The idea one which arises 
when the attention is directed to a part which results 
from the separation of an object into equal parts, is 
called a fractional unit. 

b. The idea one which arises when the attention 
is directed to a group of objects, is called a multiple 
unit. 


Remarks. 1. The,integral unit, or unit one, is the primary idea 
in arithmetic. The secondary units are definitely related to the 
integral unit by division or multiplication. 

2. An object giving rise to the idea one may be called a unit- 
object. Most writers on arithmetic call such an object a unit, 


NUMBER GENESIS. II 


3. The object which gives rise to a fractional unit is one of 
the equal parts into which the unit-object of the integral unit 
isthought asseparated In view of this relation of part to whole, 
a fractional unit is often said to be derived from the integral unit 
by the separation of the latter into equal parts. 

The wit, which is the idea one, is a unity and is incapable of 
division ; its object, only, Gan be separated, a new idea one aris- 
ing when a part instead of a whole engages attention. [See 
unity in Fleming’s Vocabulary of Philosophy. ] 

3. ANumber. A number may be defined as a unit 
or group of like units thought together. 

Remarks. 1. The units constituting a number may be integral, 
fractional or multiple 

2. Aristotle did not include wnity in the idea of number. He 

considered unity as the clement of number. Locke included 
unity in the idea of number ‘This view is adopted by modern 
writers on arithmetic. 
4. Arithmetic, asa science, exhibits the facts of num- 
ber, presents the relations sustained among the facts, 
formulates these relations into principles which bring 
the facts together until they appear in consciousness 
systematically arranged as an organic whole. Arith- 
metic, as an art, is the application of the principles of 
the science in computation. 


References on Number Genesis. 


Dr. McCosz: Wesecm to derive our knowledge of 
number from our cognition of being, and especially 
from our cognition of self as a person. We know self 
as one object; we also know other and external objects 
as singulars. Already then have we number in the 
concrete. Every object known, and especially self, is 
known asone. Every other object known, is known as 
another. one. Ii we know self as one, then the external 


bed NUMBER GENESIS. 


object which is known as different from self, is known 
asasecond one. The mind can now think of one object 
plus another one object, or of two, and of one object plus 
another one object plus another one object, or of three, 
etc. It can then, by abstraction, separate the numbers 
from the objects, in order to their separate considera- 
tion. * * * * * Having obtained in this way a 
knowledge of numbers in the concrete, and numbers in 
the abstract, the mind is prepared to discover relations 
among numbers. 


WHEWELL: The conception of number appears to 
require the exercise of the sense of succession. At first 
sight, indeed, we seem to apprehend number without 
any act of memory, or-any reference to time; for exam- 
ple, we look at a horse, and see that his legs are four, 
and this we seem to do at once without reckoning them. 
But it is not difficult to see that this seeming instan- 
taneousness of the perception of small numbers is an 
illusion. Thisresembles the many other casesin which 
we perform short and easy acts so rapidly and famil- 
iarly that we are unconscious of them, as in the acts of 
seeing, and of articulating our words. And this is the 
more manifest, since we begin our acquaintance with 
number by counting even the smallest numbers, * * * | 
We may conclude, therefore, that when we appear to 
catch a small number by a single glance of the eye, we 
do, in fact, count the units of it in a regular, though 
very brief succession. To count requires an act of 
memory; of this we are sensible, when we count very 
slowly, as when we reckon the strokes of the clock ; for 
in such case we may forget in the interval of the strokes 
and miscount. 

Now the nature of the process of counting is the same 


NUMBER GENESIS. 13 


whether we count fast or slow. There is no definite 
speed of reckoning at which the faculties involved are 
changed, and therefore memory, which is requisite in 
some Cases must be soinall. * * * * * * # 
If any one holds that number is apprehended by a 
direct act of intuition, as space and time are, this view 
will not disturb the other doctrines delivered above. 


Porter: The relation of number requires that 
objects should first be connected as wholes and parts 
and then contemplated in an arrangement which 
depends entirely upon the time relations of the mind . 
that views them. Time is the necessary condition to 
the continued subjective act of the mind in connecting 
objects into a series, and to the arranging of them as 
wholes and parts. In other words: to the act of count- 
ing, time must be assumed as both the subjective and 
objective condition; but the relations by which objects 
are viewed or connected in the act of counting when 
abstracted, generalized, imaged and symbolized, are the 
relations of number. These relations can be applied 
to material objects, to spiritual objects, to acts or states 
of the mind itself, to the very acts of the mind in num- 
bering, in short to any objects whatever, whether of 
direct or reflex cognition. 

The principal concepts of number are the wnit, the 
sum, the difference, the multiple, the divisor, (factor) and 
the ratio. These concepts cannot be defined so readily 
as they can be imaged and exemplified. They assume 
that the mind can perform certain thought-processes 
which result in certain thought-products. The psycho- 
logical condition of these processes is the arrangement of 
objects in a series. Théir logica/ condition is the rela- 
tion of the time relations, and of time itself in making 


14 NUMBER GENESIS. 


these relations possible. That number implies time, 
is further obvious from the language which we contin- 
ually use in our definitions and analyses. Wesay add 
this so many times; two times ten is twenty ; ten divided 
one time by two equals five, etc. * * * * * The 
concepts of number and magnitude are all generalized 
from the relations of concrete objects and events to 
both space and time. 

Bascom: The idea of number, like that of existence, 
is so simple and direct, so constantly merged in the 
very perception to which it is attached, as to have made 
but slight claims for explanation on sensualistic schools 
of philosophy. I see one apple, I hear several sounds. 
I feel three direct points, are examples of familiar 
expressions. The numerical notion is brought to the 
mass of colors before us as one of the waysin which 
the mind may regard it. Indeed, the same object dif- 
ferently conte: p ated, yields a great variety of numer- 
ical relations, the sensations remaining exactly the same. 
It may present several colors, and while, therefore, we 
call it one in cohesive connection, we may separate it 
into a multiplicity of parts by diversity of shades, or 
by relative position. The mind plays upon it with 
standards of its own, divides it with various linear or 
solid measurements, finds with each a diverse numerical 
expression, and terms it now one, now many, as suits. 
the purposes of thought. 

There is no object of sense which is notin some rela- 
tion one, as a tree, a grove, a forest, a world, a universe; 
and none which may not be divided and thus yield 
plurality. * * * * * The separable character of 
number from the objects perceived, is seen in the fact 
that numbers are treated independently—are accumu- 
lated in amounts entirely beyond experience, and are 


‘NUMBER GENESIS. 15 


divided and compounded by processes not founded on 
observation, or proved by it, but which belong to the 
necessary character of numerical conceptions. Our 
powerful algebraic solvents are general formulx, are 
wrought out wholly independently of things, and are 
brought to explain outside facts otherwise numerically 
unintelligible. Thus most evident is it, that in the 
most abstruse application of numbers, phenomena 
receive their solution from numerical conceptions, and 
do not, through the senses, yield it. * * * * * * 

BHxistence and number are among the most general 
of our notions, finding inherent, and toa rational mind, 
necessary application everywhere. 

Upnam: As the idea of unity is one of the simplest, 
so it is one of the earliest notions which men have. It 
‘originates in the same way, and very nearly at the 
same time, with the notions of existence, self-existence, 
personal identity, and the ike. When aman hasa 
notion of himself, he evidently does not think of him- 
self as two, three, or a dozen men, but as one. As soon 
as he is able to think of himself as distinct from his 
neighbor, so soon does he form the notion of unity. It 
exists as distinct in his mind as the idea of his own 
existence does; and arises there immediately successive 
to that idea, because it is impossible, in the nature of 
things, that he should have a notion of himself as a 
two-fold or divided person. Unity is the fundamental 
element of all enumeration. By the repetition of this 
element, we are able to form numbers to any extent. 
These numbers may be combined among themselves, 
and employed merely as expressive of mutual relations, 
or we may apply them, if we choose, to all external 
objects whatever, to which we are able to give a com- 
mon name. ; 


16 NUMBER CLASSIFICATION. 


Locke: Every object our senses are employed about, 
every ideain our understandings, every thought of our 
minds, brings this idea of one along with it. | 

BuFrFieR: The knowledge that J exist, I am, I think, 
includes this,— I am one. 

Descartes: Number is perceived by all of us in all 
perceptions of body. 

ARISTOTLE: Each sense perceives unity. 


CHAPTER IL 
NUMBER CLASSIFICATION. 


5. Integers and Fractions. 

1. On the basis of integral or fractional units used 
in their formation, numbers are classified as integers 
and fractions. 

2. AN InrecEeR. A number composed of one or more 
integral units is called an integer. 

do. A Fraction. A number composed of one or more 
fractional units is called a fraction. 


Remark. An integer and a fraction thought as combined are 
together called a mized number. 


6. Abstract and Concrete. 
1. On the basis of application to objects, numbers 
are classified as Abstract and Concrete. 


2. AbsTRAcT. A number thought as independent of, 
or separate from an object, is called an abstract num- 
ber. 

3. CONCRETE. A number thought as applied to an 
object is called a concrete number. Examples: 4 men, 
12 horses, ? of a dollar, ete. 


Remark. In the examples, 4,12 and §, are concrete numpers 
because their objects are named. 


] . va Lee Le (ih 1 
\ 


\ 
\ 


x 


\ NUMBER CLASSIFICATION, 17 


7. Composite and Prime. 

1. On the basis of divisibility, abstract integers are 
classified as Composite and Prime. 

2. CompositE. A number that can be divided into 
equal integers each greater than one, is a composite 
number. 

do. Primg. A number that cannot be divided into 
equal integers each greater than one, 1s a prime num- 
ber. 

8. Simple and Denominate. 


1. On the basis of distinct or assumed unit-object, 
concrete numbers are classified as Simple and Denom- 
inate. 

2. SimpLE. A number whose unit-object is a dis- 
tinct whole, is a simple number. As 5 books, 3 pens, 7 
horses ; the unit-objects being the distinct wholes, book, 
pen and horse, respectively. 

Remark. An abstract number is often called a simple num- 
ber. 

3. DENOMINATE. A number whose unit-object is an 
assumed time, extent or degree of intensity, is a denom- 
inate number. As 38 days, 5 feet, 7 pounds; the unit- 
objects being the day, the foot and the pound, respect- 
tively, each of which is an assumed unit-object. 

4, Compound. Two or more denominate numbers 
having the same primary unit, are together called a 
compound number. 

Remarks. 1. A denominate number is often defined as a 
number whose unit (object) is named, but such definition is 
applicable to all concrete numbers and is, therefore, too inclu- 


sive. 
2. A compound number is often defined as a number con- 


sisting of two or more denominations. Under this definition 
5 ft. 3lb. 3hr.is a compound number. The error in the defi- 
nition is apparent. ~ 


18 NUMBER CLASSIFICATION. 


9. Positive and Negative. 


1. On the basis of oppositeness in character, num- 
bers are said to be positive or negative. 

2. PostrivE. A number is said to be positiveif upon 
being applied (added) to another number it augments 
the value of that other. 

3. Negative. A number issaid to be negative if upon 
being applied (added) to another number it diminishes 
the value of that other. 


Remarks. 1. The definitions of positive and negative given 
above, are not to be considered as embracing the entire content 
of those terms as used in number science, but only as indicating 
the primary ideas attached to the terms when used indepen- 
dently of each other. More strictly speaking, they are relative 
terms, each meaning the opposite of the other; e. g., if up be 
thonght as positive, down is thought as negative; if a man’s | 
assets be thought as positive, his liabilities are thought as nega- 
tive,etc. Anumber is the negative of its opposite; i.e., while 
negative three is the negative of positive three, positive three is 
is no less the negative of negative three. Each may be called 
the positive of itself. 

2. Professor Olney is authority for the statement that it 
is unphilosophical to think of a negative number as being 
less than nothing. It is simply a number used in an opposite 
manner from some other number, and primarily belongs to the 
field of concrete numbers only. 


10. The Symbols + and —. 


As symbols of operation, the signs + and —, denote 
respectively, addition and subtraction, but as symbols 
of nature [Character] those signs denote, respectively, 
opposite qualities, conditions, or states, or opposite 
motions, directions, or tendencies; but which is to be 
regarded positive is conventional. By common consent 
a positive quantity is denoted by +, and a negative 
quantity by —. [ScHUYLER. ] 


CHAPTER IIL. 


NUMBER REPRESENTATION. 


Notation. 


il. Definition. Notation is a systematic method of 
representing numbers by symbols. 


Kinds of Notation. 


Remark, Many kinds of notation have been in use in differ- 
ent times and places. But two of these, however, are usually 
presented in arithmetic, viz.: that used by the ancient Romans 
and that introduced into Europe by the Arabs. 


The Roman Notation. 


12. Characters. The alphabet of the Roman nota- 
tion consists of seven letters, viz: I, V, X, L, C, 
D,M. 
13. Signification, I represents one, V five, X ten, L 
fifty, C one hundred, D five hundred, M one thousand. 
14. Relation. V represents five time I. 

D4 Ni: LAO cop WW 

L ne AVOu ite nk 

C of two )iL, 

D i fiver). C; 

M ‘f two “ D. 


15. Limit. The Roman notation is limited to the 


20 NUMBER REPRESENTATION. 


representation of integers, and is chiefly used in num- 
bering chapters, headings and divisions in books and 
' papers. 


16. Principles, 


Remark. As now used the Roman notation is effected in 
accordance with the following principles: 


I. Ifa letter be written with its equal or with a 
combination of its equals, the letters combined repre- 
sent a value equal to the sum represented by the let- 
ters. 


Il. If a letter be placed at the left of a letter repre- 
senting a greater value, the letters combined represent 
the difference between the values represented by the 
letters taken separately. : 


III. If a letter or combination of letters be placed 
at the right of a letter representing a greater value, the 
letters combined represent the sum of the two values. 


IV. Ikfadash be placed over a ietter or combination 
of letteys the value represented is multiphed by one 
thousand. 


17. Exercises. Read each of the following: IX; XIV; 
XXVIT;LIII; XCV; CXXVIII; XII; XCVI; MDCCL; 
MCI: MDCCL; LX XXIII; DIV. 


Write in the Roman notation: ‘Twenty-nine; thirty- 
three; fourteen; one hundred six; fifty-six; one thou- 
sand two hundred sixty-four; seventy-eight; one thou- 
sand seven hundred seventy-six; ninety-seven; forty- 
nine; five hundred thirteen; eleven hundred seventy.- - 
[Give additional exercises. ] 


NUMBER REPRESENTATION, 21 


The Arabic Notation. 


18. Characters. The alphabet of the Arabic notation 
consists of the following ten characters called figures, 
Mie 20,4, 5, 6.7.8, 9,.0. 

19. Signification. The first nine of these figures rep- 
resent one, two, three, etc., units to nine. They are 
called significant figures, or digits because they signify 
or point out numbers. The tenth figure is called zero 
or nought. It expresses no numerical value. 

Remark. In Algebra the zero is classified among the symbols 
of quantity and is defined as the representative of an infinitely 
smail quantity. 

20. Systems. The Arabic figures are used in three 
distinct systems of numbers, viz.: (1.) The decimal, 
(2.) The fractional. (8.) The compound. 


1—The Decimal System. 


21. A Scale. The series of units constituting the 
basis of a system of numbers is called a scale. 


22. The Decimal Scale. The decimal scale is a 
series of units each of which is ten times as great as the 
one next below it in the series. 


1. Ten units of the first, or units’ order are together 
ealled a unit of the second order. 

2. Ten units of the second, or tens’ order are together 
called a unit of the third order. 

8. Ten units of the third, or hundreds’ order are 
together called a unit of the fourth order, etc., etc. 

Remark. Higher orders of units are formed by thinking 


together ten units, respectively, of the fourth, fifth, sixth, sey- 
enth, eighth, ninth, tenth, etc., orders indefinitely. 


s 


22 NUMBER REPRESENTATION. 


23. Periods. 1. Thethree orders of units, viz.: units’, 
tens’ and hwndrecs’, constitute a period called wnits’ 
period. 

2. The three orders of units next higher than units’ 
period, constitute thousands’ period. Units of thou- 
sands, tens of thousands and /undreds of thousands are 
thought as composing this period. 

3. The three orders of units next higher than thou- 
sands’ period constitute millions’ period. Unitsof mill- 
ions, tens, of millions and hundreds of millions are 
thought as composing this period. 

4. Other periods of three orders each are formed of 
higher orders of units in the decimal scale. The pe- 
riods above those named are those of billions, trilhons, 
quadrillions, quintillions, sextillions, septillions, octill- 
ions, decillions, undecillions, ete., to infinity. 

5. Each period embraces three orders, yiz.: wnits, tens, 
and hundreds of that period. 


24. Lower Orders. 


Remarks. a. Thedecimal scale may include units lower than 
the unit one. 

b. The division of a unit into tenths, hundredths, thou- 
sandths, ten-thousandths, etc., is called a decimal division of the 
unit. 

1. Units which result from dividing the wnzt one into 
ten equal parts are called tenths, or units of tenths’ 
order. 

2. Units which result from dividing one-tenth into 
ten equal parts are called hundredths, or units of hun- 
dredths’ order. 

3. Units which result from dividing one-hundredth 
into ten equal parts are called thousandths, or units of 
thousandths’ order. 


NUMBER REPRESENTATION. 23 


4. Orders of decimal units called respectively ten- 
thousandths, hundred-thousandths, millionths,ten-millionths, 
ete., are formed by dividing into ten equal parts the unit 
next higher in scale. 

25. Principle of the Decimal Scale. Ten units of any 
order make a unit of the next higher order. 


Wotation of Numbers Thought in the Decimal Scale. 


26. Characters. The Arabic characters already given, 
together with a point or period, called the decimal 
point, are used in representing numbers thought in the 
decimal scale. 

27. Signification. The digits and the zero have the 
signification already stated, while the decimal point is 
used to mark the place in a written number from which 
to start in determining its value. 

28. Units’ Place. The first place at the left of the 
decimal point is units’ place. 

Remark. Any number of first order units from 1 to 9, may 
_ be represented by writing the proper figure in units’ place. 

29. Tens’ Place. The place next at the left of units’ 
place, is tens’ place. 

Remark. Any number of second order units from 1 to 9, may 
be represented by writing the proper figure in tens’ place. 

30. Hundreds’ Place. The place next at the left of 
tens’ place is hundreds’ place. 

Remark. Any number of third order units from 1 to 9, may 
be represented by writing the proper figure in hundreds’ place. 
31. Units of thousands, tens of thousands, and hun- 
dreds of thousands, respectively, may be represented in 
-the next three places at the left of those already named. 


24 ‘ NUMBER REDUCTION, . 


In the next three places may be represented, respect- 
ively, units of millions, tens of millions and hundreds 
of millions, ete. 


32. Tenths, hundredths, cnadeamaeue etc., may be rep- 
resented by figures written at the right of units’ place 
in places corresponding to tens’, hundreds’, thousands’, 
etc., at the left. 


33. Representative Scale. The series of successive 
places in which the different orders of units are repre- 
sented may be called the representative scale. 

Remark. It is called representetive, because in it the values of 


the various parts of a number are represented, and to distin- 
guish it from the thought scale —[ Art. 21.] 


Figure Values. 


[Norr. The thought of the following treatment of “ Figure 
Values” is not claimed by the writer of this book. ] 


34. The Form Value of a Figure. 
The simple, or form value of a figure is the value 
indicated by its form alone: as— 
1 indicates one first order unit. 
2 % tye iS ee 
5 ec five ‘¢ 66 6 ete. 
35. The Place Value of a figure. 


The local, or place value of a figure is the value it rep" 


resents because of its place in the decimal representa- 
tive scale. 


A figure in units’ place may be said to have no 
place value, since its form indicates its entire value. If 
1 is in tens’ place, it represents ten first order units; 
one of those units being indicated by the form, 1, while 
the other nine are indicated by 1 because of its placed in 


NUMBER REPRESENTATION. 25 


the scale. 2 in 20 represents twenty first order units; 
two of those units constitute the form value while the 
other ezghteen constitute the place value of 2 in 20. 


The place value of 3 in 30 is 27. Of 8in 80 is 72. Of 
1 in 100 is 99. Of 6 in 600 is 594, ete. 


It may thus be seen that the entire value of a figure in 
the decimal scale is made up of two values combined, 
viz.: the form value and the place value of the figure:— 
Hence the place value of a figure is the number which 
must be added to its form value that the sum shall be 
its entire value. 


The place value of a figure at the right of the decimal 
point isa negative number;e. g., the entire value of 
3 in .3 is three-tenths of the integral unit: the form 
value of 8 in .3 is three integral units, hence its place 
value must be minus 2.7, since minus 2.7 added to 3 
equals .3. 


36. Reading. Naming tke numerical value repre- 
sented by a written number is cailed reading the num- 
ber. 


Remarks. 1. In reading a number it is both convenient and 
customary to begin with the highest order of units represented, 


2. In reading a number the word end should be used only 
between units of any order and fractional units of the same 
order: a8, 325 is read three hundred twenty-five; 4.5 and .033 
are read four and five-tenths, and three and two-thirds hun- 
dredths, respectively. Such forms as 3.03 and 6.003 some- 
times occur. The former should be read, three units and three. 
. fourths of a tenth, and the latter, six pie and five- thirde 

hundredths. 


3 


20 NUMBER REPRESENTATION. 


Hxercises. 


Read 46; 326; 460; 2346; 1785; 57689; 32567; 
4567890; 45678834; 456784352; 46789726; 456.3; 3.4; 
51.6; 317.04; 456.07; 5678.17; 45.041; 32.117; 2.3456; 
4.055 ; 56.78946; .346; .3; .5678; .56789047; 03456789 ; 
45678 ; .5678789. 

Write each of the following in the decimal scale :— 
Two hundred thirty-four; seven thousand sixty-five; 
Five hundred forty-one; seven thousand seventy-six ; 
two thousand ninety-eight; four hundred thousand 
sixteen; forty-five thousand ten; sixty-six thousand 
ninety-four; seventeen thousand five; nineteen thou- 
sand nineteen ; five hundred thousand six ; six million 
four hundred thousand seventy-eight; three hundred 
million-seven thousand six hundred nine ; three tenths; 
nine tenths; fifteen hundredths; seven hundredths; 
fourteen thousandths; six thousandths; two ten-thou- 
sandths; three hundred eleven thousandths ; one hun- 
dred two ten-thousands; twenty-four hundred thou-~ 
sandths; five hundred and seven hundred-thousandths; 
sixteen millionths; four hundred'sixty-one millionths ; 
fifty-seven ten-millionths; fifteen tenths; twenty-one 
tenths; two hundred thirteen tenths; five hundred 
tenths; seventy tenths; one thousand seventy-five — 
tenths; twenty tenths; one hundred fifteen hundredths; 
two hundred six hundredths; four hundred forty-one 
hundredths; fifty hundredths ; six hundred hundredths; 
four hundred fifty hundredths; four hundred seventeen 
tenths ; three thousand hundredths; four thousand two 
hundred eighty-four thousandths; two thousand seven 
thousandths ; one hundred tenths; forty-five thousand 
thousandths; forty tenths; ten thousand ten-thou- 
sandths ; two hundred and six-tenths. 


NUMBER REPRESENTATION. | 27 


3.—The Fractional System. 


37. The fractional system has for its basis the series 
of fractional units, viz :—4, 4, t, 4, 4, 4, 4, etc., to infin- 
ity. 

Any one of these units equals the unit next lower 
increased by the part of the lower that the higher is of 
the unit 1; e.g, 3=$+4 of 4; 3=1+4 of 4; 4=444 
of $, ete. 

38. Two numbers are necessary to the conception of a 
fraction. a. The number of equal parts into which 
the unit is thought as separated. 6. The number of 
fractional units that are thought as constituting the 
fraction. 

39. These numbers are called the terms of the 
fraction. The number of equal parts into which the 
unit, or whole, is thought as separated is called the 
denominator of the fraction. The number of fractional 
units thought as constituting the fraction is called the 
numerator of the fraction. 


40. Denomination. The fractional denomination of 
a fraction is the same as the ordinal of the denomina- 
tor. This is true of all fractions except those having 
the numbertwo fora denominator. The denomination 
of such fractions is Aalf instead of second, the ordinal of 
the denominator. : 


The Notation of a Fraction. 


A4l.. Since two numbers are necessary to the thought 
fraction, two written numbers are necessary to notate 
.a fraction. These written numbers have the same 
names, respectively, as the terms of the thought fraction 
which they represent. 


28 NUMBER REPRESENTATION. 


42, The written denominator is placed below the 
written numerator and separated from it by a short 


line. 

Remarks, 1, A fraction is thought as sustaining a definite rela- 
tion to the integral unit; we, therefore, think of a written frac- 
tion as attached to units’ place in the (representative) decimal 
scale. If the thought fraction require it, the written fraction 
may be attached to any other place in the representative scale. 

2. A fraction which results from a decimal division of the 
unit may be written in the decimal (representative) scale. 


[Exercise in writing and reading fractions. | 


3.—The Compound System. 


Remark. Under what is here called the compound system of 
numbers are included all denominate numbers. Since each 
attribute measured has its own distinct scale of units, it would, 
perhaps, be better to speak of the compound systems (instead of 
system) of numbers. 


43. A compound number is thought in two or more 
different orders of units that have the same primary, or 
standard unit. 

44. In compound numbers the number of orders 
(denominations) in any ‘“‘measure” is limited to the 
number of different unit-objects agreed upon for meas- 
uring the attribute under consideration. 


45. The compound system of numbers is based, not 
so much on the fact that the several scales are varying, 
as that each ‘‘ measure” has its own scale. Some of 
these scales are varying and some of them are uniform. 
fach of the common measures has a varying scale, 
while the “‘ metric” measures, including the measure 
of U. S. money, has a uniform and decimal scale. In 
the old books will be found a duo-decimal ‘‘ measure” 
which is, of course, uniform. 


NUMBER REDUCTION. 29 


46. Each denominate number which forms part of a 
compound number, is thought in the decimal system, 
in the fractional system, or in both. 

The Netation of a Compound Number. 
47. The denominate numbers which compose a com- 
pound number, are written in a descending series, from 
left to right. Thus—4 bu. 3 pk. 5 qt. 1 pt.; 5 da. 16 hr. 
47 min.; # lb. 34 oz. $ pwt. 
48. The names of the orders or denominations may 
be abbreviated, but the parts of a written: compound 
number are not to be separated by any mark of punc- 
tuation. 

In reading acompound number the word and should 
not be used between any two adjacent denominate 
numbers composing the compound number. 

Remark. The first compound number given under Art. 47, 


should be read—4 bushels 3 pecks 5 quarts 1 pint. The third 
should be read—# of a pound 3} ounces # of a pennyweight. 


[Exercise in writing and reading compound num- 


bers. | 


CHAPTER IV. 


NUMBER REDUCTION. 
49. Reduction consists in the change by which a given 
numerical value is thought in another order or denom- 
jnation. 


50. Reduction Descending. Reduction descending 
consists in reducing a numerical value of any order or 
denomination to a lower order or denomination. Itis 
effected by thinking the value of each unit of the 
given order or denomination in the number of units of 
the lower that are together equal to a unit of the 
higher. 


30 NUMBER REDUCTION. 


51. Reduction Ascending. Reduction ascending con- 
sists in reducing a numerical value of any order or 
denomination toa higher order ordenomination. Itis 
effected by thinking as a unit of the higher the num- 
ber of units of the lower order or denomination that 
are together equal to a unit of the higher, 


Exercises. 


REDUCTION DESCENDING, 


52. Casel. Of numbers thought in the decimal system. 


Remarks. 1. Thesigns*.: and .*. are convenient to use in some 
of the written forms. The former is read “since” or “because,” 
and the latter is read “‘ hence ”’ or “ therefore.” 

2. In many of the exercises that follow, u is used for units, 
t for tens, h for hundreds, th for thousands, tth for tenon! 

sands, .t for tenths, .h for hundredths, .th for thousandths, .tth 
for ten-thousandths, ete. 


Example. Reduce 9 tens to units, 
First form. 


°° 1 ten=10 times 1 unit, 
9 tens=10 times 9 units. 
10 times 9 units=90 units. 
.. 9 tens=90 units. 


Second form. 


*.° 1 ten=10 units, 

9 tens=9 times 10 units. 

9 times 10 units=90 units, 
.. 9 tens=90 units. 


53. 


1 
2 
3 
4 
5 
6 
7 
8 
9. 
10. 
11 
12 
13 
14 
15 
16 


NUMBER REDUCTION. 31 


. Reduce 2 tensto units. 17. Reduce .03 to tenths. 
(74 ¢¢ a4 ¢ 18. cc 04 (79 cc 


6c 90 6c 6c 66 39. off 43 ¢¢ 74 
Case Il. Of numbers thought in the fractional system. 
Example. Reduce 2 to 15ths. 


First form. 
pk Lata 1 
. +=5 times ;,, 
Pe By 3 
3—3 times ;%;. 
3 times 73; "5. 
ae ee 
CS Seine HX 
Second form. 
sce la 8 
oR A Bo 
soit 3 3 
3—3 times ;%. 
3 times ~-—,*s. 
‘51 


Remark. If the relation of equality between 4 and 


33; be not readily seen, the above solution may begin— 


1 


2. 


1=t, $=$ of T or 15) ete. 
. ReduceZ to 6ths. 4, Reduce # to 30ths. 
meee -8ths, 5. 6S & ~40ths. 


8 « §& “ 18ths. 6. “£6 O5ths. 


32 


7. Reduce # to 20ths, 


8. than pAtLB, Le “§ 
9. Si Bi Aths, 18, h 
10. So eto ths. 19. 
11, Tea Sa as. 20. ‘ 
12. ec 6S SO 1sts. 21. } 
13, se ei Othe: 22, , 
14, Oe ys are, 23. e 
15. Cb ae aOLDS 24. sh 
54. Case Ill. Of numbers thought 
systems. 
Example. Reduce 3 pk. to qt. 
First form. 
*’ 1 pk.=8 times 1 qt., 
3 pk.=8 times 3 qt. 
8 times 3 qt.=24 qt. 
“od pk.=24 qt. 
Second form. 
eo Deena tee 
3 pk.=3 times 8 qt. 
3 times 8 gt.=24 at. 
| ° oO pk, ==24 qt. 
1. Reduce 3 bu. to pk. 
2. 2 pk. to qt. 14. 
3. HOO GLE be 15. eS 
4, hae path ae id as 16. + 
D. Peco We Kerb, Cee wiki eS 
6. SS on. Aeon BL ete. ¥ 
as Semin, to. seca’ a1Y: 3 
8. ee LV, LOL, 20. i 
o ha cneyit te Orman: 21. “k 
10. Po ie ah Sibert g Bes 22. . 
LS Ce pw. toner: By . 
12. “eC Aoueal. toed, 24. . 


NUMBER REDUCTION, 


16. Reduce 4to 9ths. 


3 “ Q4ths, 
$ “ 12ths. 
+ “ 28ths. 


in the Connaneiea 


13: Reduce 5 wk. to da. . 


CO OU Cn DE Cho D~IOd 


55. Case I. 
Example. 


NUMBER REDUCTION. 


33 


REDUCTION ASCENDING. 


Of numbers thought in the decimal system. 
Reduce 60 units to tens. 
First form. 
. lu, of 1t, 
60 u = +), of 60 t. 
js of 60 t=—6 t. 
etoU wie 6 t. 


Second form. 
Mba lA bah eres Gat 
60 u = as many t as 60 
is times 10. 
60 = 6 times 10. 


OU == 6 
1. Reduce 40 uto tens. 13 Reduce 40 tenths to u. 
ay: 6é 7 Ce 66 6s ’ 66 60 66 cc 6 
3. 66 20 66 66 e 15. 66 90 66 66 66 
4. o¢ 90 6c 66 6“ 16: 66 100 66 6c GG 
5. 66 30 t “ h, LF, 6c 110 66 6c 6 
6. é¢ 50 CC 66 6e dee 66 30 ch (6 t 
if iT 70 ee a3 66 19: 4 50 66 ae 5 
8. cc 90 ee 06 66 20. “ 70 “¢ in 14 
9, 6 100 6 66 6é 21. 6< 100 é 4 6c 66 
10. JE eh ea 22. HOU wy abbey or cocks 
1B pmo be Ps, vty) 123: has 8 9b NV te a Aad 
12. 66 70 66 66 ¢é A | 6% 40 46 cc GS 
“56. Case Il. Of numbers thought in the fractional system. 


Example. 


Reduce 3% to Sths. 


Lirst form. 


34 NUMBER REDUCTION. 


Second form. 


7s =as many dths as 
9 is times 3. 
9 is 3 times 3. 


8 zs = %. 

. Reduce 4 to 3ds. 11. Reduce 38, to 5ths. 

: 8) te 4ths, 12. 6 Sr eee 
Be DOS. 13. 42  8ths. 
«yy “ halves. . 14. ho Sas. 
«fs “ Sths. 15. «42 “Sths. 
“2; “ 10ths: 16, fe “ Tths. 
ce Ag “8 ds. 17. cfs “ 25ths. 
642“ Aths. 18. “343 “ *1dths. 
feet Othe: 19, «of “ Tths. 
e432 Oths. 20. 46 “ D0ths, 


Case II. Of numbers thought in Compound systems. 


Exaunpie. Reduce 20 pk. to bu. 
First form. 
‘ud (pk Oreboous 
20 pk. =} of 20 bu. 
+ of 20 bu. = 5.-bu: 
. 20 pk. =5 bu. 


Second form. 


as AAI eee 
20 pk.==as many bu. as 
20 is times 4. 
20 is 5 times 4, 
BU. Dk c= One 


NUMBER PROCESSES. 35 


1. Reduce 6 pt. to qt. 11. eae 36 3 to th, 

2. fimo pK. to, buys! 12. 4 tO ys 

3. ‘¢ 24 qt. to pk. 13. oe yy A Ea) DUO os 

4 “12 ft. to yd. 14. “ 28 da. to wk. 

5 30 ft. to yd. 15... °.“* 48 Troy oz, tolb. 


6 13 ft. to yd. 16. «64 Av. oz. to Ib. 
ip ¢ 24 in. to ft. WE 68 gal.-to bbl. 
8. ‘48 in. to ft. 18. ¢ 300 lb. to cwt. 
9. (2 in. to ft. 19: «¢ 6480 sq. rd. to A. 
10. “48 gr. to pwt. 20. 640 sq. rd. to A. 


CHAPTER V. 


THE NUMBER PROCESSES, 


58. Preliminary Note. In the early stages of mind 
growth, number is considered in connection with the 
objects in which itis found. In fact the child mind 
does not clearly discriminate between visible and tan- 
gible objects and their number attribute. Quite early, 
however, the mind becomes sufficiently reflective to 
deal with number apart from the concrete phase in 
which it is first considered. The number processes of 
addition, subtraction, multiplication and division, are 
then found to consist, not in counting objects together, 
nor in counting them apart, but in recalling from mem- 
ory, a number (in any case) that bears a certain re- 
quired relation to one or more of the numbers given. 
IZlustration: ‘The process indicated in each of the fol- 


30 NUMBER PROCESSES. 


lowing consists in giving, at once from memory, the 
number which completes the equation, 


1 5+4=> 
2. 12—7= 
D5) nO ae 
4. 10+2= 
By a aibes 


As the mind progresses in its development, the less 
does it recognize the distinctive synthetic and analytic 
phases assumed and necessitated in the primary and 
concrete aspect of number work, and the more does it 
come to comprehend that the number processes consist 
in the perception of numerical relations and the num- 
bers which answer to those relations. | 

[Read Everett’s Science of Thought, pages 102 and 
103, et seq.] 


Remark. The number processes are first performed in com- 
bining and in separating objects, which to the child, stand for 
numbers. The processes may, therefore, be classified as syn- 
thetic and analytic. 


The Synthetic Processes. 
I.—ADDITION. 


59. Sum. The sum of numbers is a number equalin 
value to those numbers thought together. 


Remark. Iitwo numbers are of like character (i. e., both positive 
or both negative) their sum contains as many units as the num- 
bers added. If, however, the numbers to be added are of 
unlike character (i. e., one positive and the other negative) 
their sum contains as many units as the difference between the 
absolute values of the two numbers; e. g., the sum of plus 4 and 
minus 3 is one. 


ADDITION. 37 


60. Addition, Finding the sum of numbers is called 
addition. 


61. Addends. The numbers to be added are called 
addends, 


62. The Mental Acts Involved, 


a. The mind perceives the addends. 
b. It compares them in respect of denomination 
and order. 


c. It immediately recalls from memory the sum — 
of the given addends. 


Remarks. 1. The third act involved in addition presupposes 
a preliminary stage of work upon which this act is based. In 
the preliminary stage which leads up to addition, the mind per- 
ceives and compares, as stated underaand b. It then begins 
with the number of units in one of the addends and counts 
until the units of the other addend are used. The number 
reached is the sum of the two addends. 


2. With the sum of two numbers obtained as above stated, 
the mind may combine another number, and so may continue 
the act of combining a new addend with the sum already found 
so long as there are numbers to be added in a given case. 


63. Reduction. 1. In adding numbers of any order 
in the decimal scale, a sum exceeding 9 is often obtained. 
Since such sum cannot be represented in the place 
which corresponds to the order of units composing the 
sum, a part or all of thissum must be reduced to units 
of one or more higher orders before it can be notated 


2. In adding two or more fractions a sum is often 
found which equals or exceeds a unit of a higher order 
(either fractional or the unit 1.) A part or all of this 
sum may be reduced to units of a higher order before 
expressing the ultimate sum, 


38 NUMBER PROCESSES. 


3. In adding several denominate numbers of the 
same denomination a sum is often obtained which 
equals or exceeds the number of units of that denomi- 
nation that are together equal to a unit of the next 
higher denomination A part or all of this sum may 
be reduced to units of a higher denomination before 
expressing the ultimate sum. 


64. The Sign. A perpendicular cross (+) placed 
between two symbols of number indicates that the 
numbers represented are to be added together. ‘The 
sign is read plus. 


65. Principles. 1. Only like numbers can be added 
together. 


Remarks. 1. Like numbers are abstract numbers (either inte- 
gral or fractional) having like units, or such numbers made 
concrete by having like unit-objects. 


2. If unlike numbers are to be added together they must be 
reduced to the same name or order. They may then be thought 
together under that name. 


II. If the addends be used in parts the sum of the 
partial sums thus obtained equals the sum of the 
addends as wholes. | 


Exercises. 


66. Addition of Decimal Numbers. 


if 
1. Begin with 1 and add by3’s to the sum nearest 70. 
9. ce 1 6 A’s ce 70. 
3. if 1 pp 7 70. 
4 <4 1 6 6’s ' 4 70. 


ADDITION, 39 


5. Begin with1 andadd by 7’s to the sum nearest 70 
6. sf 1 ent f i 70. 
7. . 1 Meir Og PE 70. 
8. 4 1 Stas x 70. 
9; *S 1 pone Uh a 70. 
10. rt 1 ial as ¢ 70. 
rt . 2 aedno os ss 70. 
12. 2 ie 4 70. 


[Continue until the pupils have used every integral 
number from 1 to 12 as a beginning addend and have 
added constantly each number from 3 to 12 to the pre- 
scribed limit or to any other limit that may be desig- 
nated. | 


EY. 
Find the sum of each of the following: 


(11) (42) (18) (44) (15) (46) 47) 
456+123-4921429111924-129-+345, 
258-+589-+358-+-720-+396--461-+472, 
849-1 634-1481 1.963-+548-++878-L975. 
345-1.567-+492-1276-+4-482-1'876-L543, 
279-+614-+344-4789-+-759-+ 456-524. 
863-4 148-+-672-1 678-+.927-1259-1 356, 
546-+.927-+249-1-762-1.936-+783-+247, 
234+520-+295-+279-+143-+-429-1 642, 
998-1. 873-579-1887-1396- 674-1378, 
674--787-+918+-784+849-+469+367, 


$80 Gort G2 Ste 09: hoor 


a 


40 


pt 


— 


= 


CO. Sot CU 


0 OO ES re 


Senet 2 Se OLS Fe 


NUMBER PROCESSES, 


Tek: 

CEL) Dyes C18) CL ee 
oOo an a ped eae at ea nail aaa 

873-+82015-+-37148+ 98765+34596. 
4863.1 78056-L49354.-78¢0 LAGE 
7563-58597 +68734-+-59876-+ 25679. 
3678+ 49912-85876 -59378-+ 34567. 
1971-+-54893-+ 67854-+-98763-+56787. 
6798-+-74235-+- 78967 + 64576-+ 34568, 
9763-++-45897 +67854-+ 76345-+-56789. 
7867-+84856-+-78543 +-98675-+- 78967. 
9876-23547 -+ 87967 -+- 85456-+ 89767. 


IV. 


(11) CD) 18) a es 
8678941456235 +-473352-1546783. 
632106-+543765-+524731-+453217. 
745387 + 632873-+475269-+267831, 
254613-+367127-+314572-1572546. 
257362-+-432569-+ 685428-+. 427454, 
742.638-+567431-+482157-+632457. 
467325 +.825642-1+567843-+-367543. 
532675 +.174358-4+357862-- 156732. 
786322-+ 637852-1 642138-+. 463784. 
111865-+231124-+256782-1536216. 


V. 
(11)) (12) > (18). 4) 
3.4 169.8 WR k 56) Se 
6.7 OES 8 Be 4) ON ee 
5.07 +408 4 5.074 809) O4p aes 
12.05 +1617 +1708 +1921 + 5.18 + 2.05 


13.17 +2125 +19,02 167.14 +10,11 113.18 
24.25 +98.06 +74,01 +9321 +71.16 +62.24 
61.004-+ .006-+13.002-+13.006-- 4006-+74.112 
74.124+ 6111-118.186-+11.018-+14.016-L€5 125 
12.118-+ 7.114-+-21.267-+13.576+36.044-+76.324 
78.016-+ 7 485-+34 361-+12.679-+48,.327-191.912 


| ADDITION, 
| 67. Addition of Fractions. 
f. 


(7) (8) (9) (10) (11) (12) 
sim oe hl aon: ae MH 
$+4+E+9+844 
b+d+S+4+344 
b+e+3+4+844 
$L4+3+4+344. 
$+44+4+9+49+3 


Pe PE ee OSS ee 


-_ 
SA 
feo, 
- 


OY CokS toe Hb onfoo oot SI 
++4+44+4+ 
COR OW HI cope Co ApH OO 
++++4+4 


i 
-_— 
— 


of coho PK top oto cots CO 
+++++4+4., 
coke orks BHR WKS cob nh S 
+4++4++4+4 2 
A ops PP BE cote ee 
+++ ++ 
ence cr SY AKO pol colbo 


—_ 
— 
— 
—_—~ 
— 
bo 
— 


oe ey 


pH HDS 


(7) (8) (@) (0) dt) 2) 
Pees eet CB eo, Se 
Set ue eet lt 
eee oe sot Let YS. 
he eet eat Yet S- 
Sper vet ts ts 
tr gore de ee toe: 


OR 99 bp 


NUMBER PROCESSES. 


Addition of Compound Numbers, — 


gal. gt“ = pt, 
3 a 1 
pierce 1 
( Dorit 
4 Pais 
(3) 

Ibe one prwrtos or: 
a | 402.16 
1B We, 1625, 14 
4 8 10a 

(5) 
bu. pk. qt. 
1 ees 
3 y Aaa 5: 
1 38 4 
3 2 ao 
Fil, an 
Vl Ste eee 
1 2 Si 
3 1 fs 
1 2 4 
Liiscane 
(9) 
cu.yd. cu ft: cusin, 

1 10 14 

8 15 35 

1 20) 47 

2 40 


wk. 


Orbs 


wk. 


Ord bor 


_ 


do howe 


CO 92 Si CO =< 


da. 


H> OD 


-— 


CU Hy oo 


be 
° 


da. 


Cp UU Or 


pk. qt. 

2, 7 

3 4 

vi 5 

yA heen 

x (4) 

hr. 

3 16 
10 20 
20 30 
(6) 

oz. pwt. 
3 16 
8 15 

5 18 
¢ 10. 
(S) 

Sabb reeves 2 
2 15 
5 10 
7 13 
7 4 

(10) 

Hro nin 
15) 264 
Ts eg 

Gt aula 

21>) 10 


min. 


sec. 
15 
40 
15 


. ae ae 


Fre ene 
ines or 
Fe, OG 


kt CO DO CO 


ADDWION, 


mm Stoo yp “St 


cu. yd. 


ne 
roo bo Or S, 


momo Z 


tA 
ee 


(12 
fo2e) OTT 
Ai 9 
2 7 
18 4 
eer HH 
(14) 
pk. gq 
1 7 
7, 3 
3 7 
1 6 
(16) 
re He lb # 
2 9 
if i 
2 fs 
Peat 
(18). 
cu. ft. 
20 200 
Lo 400 
17 1600 
(20) 
fare ds 
7 15 
6 10 
5 17 


cu. in. 


44 NUMBER PROCESSES. 


II.—MULTIPLICATION. 
69. The Genesis and Nature of Multiplication. 


1. In addition the addends are compared in respect 
of unit-object Gf concrete) and order, or denomina- 
tion. They may be compared in respect of equality. 
If several equal addends are used in succession, the 
additions together may be called a constant addition. 

2 Ifthe sum ofa constant addition is recalled from 
memory when one of the equal addends and the num- 
ber of those addends are known, the act of thus recall- 
ing the sum is called multiplication. The sum thus 
found is called the product of the given addend by the 
number of those addends. The given addend is called 
the multiplicand while the number of the addends is 
called the multiplier. ; 

3. The numerical value of the product is dependent 
upon two numbers, viz:—the multiplcand and the 
multipher; hence these numbers are called factors 
fmakers} of the product. Mathematicians do not 
grant, however, that the multipher aids in making the 
product by entering into it as one of itsaddends. The 
multipher is conceived as a factor [maker] of the pro- 
duct in that it (the multipher) is the numerical rela- 
tion of the product to the multiplicand. 

4. An examination of multiplication, as it is pre- 
sented by writers on arithmetic and algebra, reveals 
the fact that the product bears the same relation to the mu!- 
tiplicand that the multiplier bears tol. Multiplieaticn 
in its fulness is, therefore, the finding of a number 
that bears the same relation to the multiplicand that 
the multiplier does to 1. The act itself is, of course, 
an act of memory—that of recalling the product when 
‘the factors are known. 


MULTIPLICATION. | 45 
4 


Formal Definition of Terms. 


70. Product. The product of twonumbers is a num- 
ber that sustains the same relation to one of them that 
the other does to 1. 


“71. Multiplication. Finding the product of numbers 
is called multiplication. 


Remark. Mathematicians put into the term multiplication a 
broader meaning than that implied by its etymology and by the 
definition usually given. The process may be the finding of a 
number of times a given number; it may be the finding of one 
or more of the equal parts of a given number; it may be the 
finding of the negative of a number of times a given number, 
it may be the finding of the negative of one or more of the 
equal parts of a number. 


72. Multiplicand. The multiplicand is usually defined 
as the number to be multiplied. 


73. Multiplier. The multiplier is usually defined as 
the number by which the multiplicand is (or is to pe 
multiplied. 


“74. A Second Definition. The multiplier is the ratio 
of the product to the multiplicand. 


Remarks. 1. The second definition herein given is believed 
to embody the full meaning which mathematicians attach to 
is term multiplier as used in arithmetic and algebra. 

. If the multiplier is an integer, the multiplicand may be 
thought as one of the equal addends of a constant addition, 
while the multiplier is the number of those addends. : 

3. If the multiplier is a fraction the process of multiplication «> 
consists in finding such part of the multiplicand as the multi- 
plier is of 1. 

4. If the multiplier is a negative number, the product is the 
negative of what it would be if the multiplier were a positive _ 
number. 


¥ rw ee Zo +” ¢ ‘dl Wd 
te ee A Pa 
: Mi < 


46 NUMBER PROCESSES. 


75. The Multiplier of Units’ Order.—If the multi- 
plicr consist of units of the first, or units’ order, the 
product is units of the order of the multiplicand; e. g., 


3 times 4 wnits = 12 wnits. 

ie Sc rd eres | Se ees 

3 “* 4 hundredths = 12 hundredths. 
3 

9 

3) 


- 
os Se 
. 


“4 tenths = 12 tenths. 


76. Of Higher Order Units. If the multiplier consist 
of units of an order higher than units’ order, the 
product is units of an order correspondingly higher 
than that of the multiplicand. Example: a. Multiply 
4 un its by 3 tens. 


Solution. 3 times 4 units = 12 units, but since the 
multiplier is of tens’ order, the product is 12 of tens’ 
order; 1 e., 12 tens. 


b. Multiply 4 tens by 3 tens. 


Solution. 3 times 4 tens = 12 tens, but since the 
multipher is of tens’ order the product is of an order 
next higher in the decimal scale than that of the mul- 
tiplicand, and hence is 12 Aundreds. 


77. Of Lower Order Units. If the multiplier con- 
sist of units of an order lower than units’ order, the 
product is units of an order correspondingly lower 
than that of the multiplicand. 


Example, a. Multiply 4 by 3 fourths. 


Solution. 3 times 4 units = 12 units, but since the 
muitiplier is of fourths’ order, the product consists of 
units ¢ as great as units of the order of the multipli- 
cand, and hence is 12 fourths. 


MULTIPLICATION, 47 


b. Multiply 4 by 3 hundredths. 

Solution. 3 times 4 units = 12 units, but since the 
multiplier is of hundredths’ order, the product con- 
sists of units .01 as great as units of the order of the 
multiplicand, and hence is 12 hundredths. 

ce. Multiply + by 3 fourths. - 

Solution. 3 times 4 fifths = 12 fifths, but since the 
multiplier is of fourths’ order, the product consists of 
units } as great as units of the order of the multipli- 
cand, and hence is 12 twentieths. 


78. Reduction. In effecting multiplication of a deci- 
mal number, a product exceeding 9 is often found. 
As such product cannot be written in the place in the 
r-presentative scale that corresponds to the order of 
units constituting the product, it becomes necessary to 
-reducesto-units of a: higher order. or orders so much 
of the product as is thus reducible without involv- 
ing fractions. 

In multiplying a fractid’ or a denominate number 
the opportunity for reduction is often presented. 
Reduction ascending is employed. 

79. The Sign. The sign of multiplication is an oblique 
cross (X) placed between the two factors. It is read 
times if the multiplier precede it, or it is read multiplied 
by if the multiplier succeed it. If the multiplier be a 
fraction not exceeding the wnt one, the sign should be 
read of instead of times if it follow the multiplier. 

“Remark. There is a lack of uniformity among teachers in 
regard to the order of arranging the written terms of an indi- 
cated multiplication ; some insisting upon placing the multipli- 
cand and some the multiplier first. In business calculations 
where rapidity is of moment, the multiplier is thought and 


spoken first. As 3 times $t = $12. In addition the number 
Wiitten before the sign is to be affected by that which follows 


48 NUMBER PROCESSES. 


jt. The same is true in an expressed subtraction and also in an 
expressed division. It would seem, therefore, that a similar 
arrangement would be logically dictated in expressing a multi- 
plication. 

80. Principles. 

I. The product is of the same denomination as the 
multiplicand. | 

Remark. This principle implies a concrete multiplicand or 
that the multiplier be considered first order units. 

II. The product is of the same order as the multi- 
plier. [Olney.] 

Remark, This principle is interpreted as meaning that if the 
multiplier be units of the first, or units’ order, the product is - 
units of the order multiplied; if the multiplier be tens, the pro- 
duct is tens of the order multiplied; if the multiplier be tenths, 
the product is tenths of the order multiplied ; .if the multiplier - 
be thirds, the product is thirds of the order multiplied. Illus- 
tration. If6 be multiplied by 2 units, 2 tens, 2 tenthsand 2 thirds, 
respectively, the several produéts are 12 units, 12 tens, 12 tenths, 
and 12 thirds. [See Arts. 75, 76 and 77.] 

III. The multipher is an abstract number. 

Remark. This principleis true in the light of the office of the 
multiplier as already stated, and as exhibited in the statements 
and usage of mathematicians. 

IV. If either factor be multiplied (or divided) the 
product is multiplied (or divided) by the same num- 
ny 7 

Mart both factors are abstract they may be used — 
inter NR a without affecting the value of. the 
produet. ; 

VI. If either’ or both factors be used in pale the 
sum of the partial products equals | the product of the 
factors as wholes. 


MULTIPLICATION. 49 


Remark. This principle is usually applied in the multiplica- 
tion of factors each of which occupies two or more orders in the 
decimal scale; e. g., if 324 is to be multiplied by 36, the steps 
are—6 times 4 units, 6 times 2 tens, 6 times 3 hun dreds, 30 (or 
3 tens) times 4 units, 30 (or 3 tens) times 2 tens, 30 (or3 tens) 
times 3 hundreds. The six partial products combined equal 
the entire product of 324 'iy 36. 

VII. The product sustains the same relation to the 
multiplicand that the multiplier does to 1. 


VIII. A fraction is multiplied by an integer by mul- 
tiplying the numerator or dividing the denominator 
by the integer. 


IX. A number is multiplied bya fraction by obtain- 
ing such part of the multiplicand as the multiplier is 
of 1. [See Prin. II. and III.] 


Exercises. 


SI. Multiplication of Decimal Numbers. 


“8 Multiply 125 by 5. 19. Multiply 1825 by 24. 
2. boat Qi" 20, 1456 15. 
3 fi Boe Pyeng 21. “ 1234 “ 36, 
4 4 492 9. 22. is 721i “ 44. 
5 oS GLO. On) Qo. Ua aay x fe DOr: 
6 if PAV 9.0, 24, Myer LODO to BU2, 
7 . Paoli bitiat ec Zoe ria ato lat S25, 
8 ns 12s 1G,” 820; Gtaue (21o ee. OgOr 
9 : ROD ys Laie tok eee ts 128.408 C8 BO 
10. bs Big oie 4.228. “8.4661 “ 225, 
ue, . ahaa On etad. wee BOOT 2. cf 200. 
12 cS Be, A NOU. He vo Nise yp ROAD. Sey Qa, 
13 ~ DEON Gta HO Le Cl 9200 “ 760. 
14 : OTD Vie 0.59) 0-2. 80005“ 444. 
15 $2238 “ 17. ‘33. Po BOLE ORS ria. 
16 s S251 “12. 34. ee. Book SS a27, 
17. “é 9213 “20. 3. 67234." S328, 


18: Be OTA SCID. OBE: 921385.“ 462. 


50 


co 
aD or WN F Pb 


sO 90 


a es ee ee 
aor WN © 


NUMBER PROCESSES. 

Multiply 100.21 by 329. 
f 222345 “ 4252, 
. 67.2131 “ 1008. 
345621," 4254, 
8.76345 * 4456, 
é 97.2341 “ 3256, 
a 100456 “ 42515, 
a 728452 “7112. 
tk 976213 “ 3452. 
¥ 4302152 “ 4670. 
F 6821345 “. 7612. 
: 8789.232 * 3462. 
4821345 “ 6722. 
a 2745672 “ 4518. 
& 6821345 “ 31245. 
is 8972112 “ 4003-4. 
6c 


1213.43 ‘6 380352. 


Multiplication of Fractions. 


16. Multiply 2? by 


Multiply 


by 


2. 


ot pbs oko ake Cp CO PP CO NTR OF HR aI OD 


1% 
18. 


“c 


$ 


ho CHR Ko DH do we ote 


as 
Ga RE ote 


Ie ole ~te 


73 


len oye OY exoe Hee ones Sie ORO 


e 


cot wm cS pha foe 420 GRO 


MULTIPLICATION, 51 
83. Multiplication of Compound Numbers. 
gal. qt. pt. gi. 
(1) Multiply 5 3 1 38 by4. i 
bu. pk. qt. 
(2) Multiply 3 2 = 3 by 5d. 
Vauctte in. 
(3) Multiply 4° 2 10 by 7. 
cu. yd. cu. ft. cu.in. 
(4) Multiply 9 13 40 by 18. 
Ib. oz. pwt. gr. 
(5) Multiply 4 3 10 10 by 1d. 
Mit are Fae 
(G) Multiply 8. 7 15 by 4. 
bu. pk. qt. 
(7) Multiply 7 2 3by8 
WON Se bon ol Th: 
(8) Multiply 4 2 9by83 
wil da.) nt, 
(9) Multiply 3 4 15 by 10. 
sq. yd. sq. ft. sq. in. 
(10) Multiply 3 4 22 by 12, 
Seo 2D ere, 
(11) Multiply 3 4 2 1 14 by8 
Wie AUT, se. 
(12) Multiply 7 6 15 by 4 
ewt. qr.° Ib 
(13) Multiply 4 3 10by8 
bu. pk. qt. 
(14) Multiply 7 3 7 by 15 
Ib. oz. pwt. gr. 
(15) Multiply 7 6 18 10by9 
gal. qt. pt. gi. 
(16) Multiply 7 2 1 3by7 
A\WE sity oe 
eynors BER , 


SL 
to 


NUMBER: PROCESSES. 


wk da hr. min. 
(17) Multiply 7 4 10 15 by 19. 


bu. pk. qt. 

(18) Multiply 8 3 2 by 6. 
IDG Sie mie 

(19) Multiply 7 3 4 1 by 5. 


lb. oz pwt. gr. 
(20) Multiply.2 11 15 10 by 8. 


ety: Than. 
(21) Multiply 4 2 9 by 7. 
bu. pk. qt. 


(22) Multiply 7 1 2 by 12. 


III—COMPOSITION. 


84. The continued product of several factors is found 
by multiplying the product of two of them bya third, 
the product of the three by a fourth, and so on until 
all the factors are used. | 

85. A Composite Number. The product of two inte- 
gral factors, each greater than 1, or the continued pro- 
duct of several such factors is a composite number. 
86. Composition. The process of forming a composite 
number is called composition. 

87. A Prime Number. A number that cannot be 
formed by composition is called a prime number. 

88. Numbers are relatively prime if they have no 
common factor. | 

89. A Common Multiple of numbers is a product of 
which each of them is a factor. ; 

90. The Least Common Multiple of numbers is the 
least product of which each of them is a factor. 

91. A Common Factor of numbers is a number that 
can be used as a factor of each of them. 


INVOLUTION. 53 


92. The Greatest Common Factor of numbers is the 
greatest number that can be used as a factor of each of 
them. 


93. Principles. 


I. Both prime and composite factors may be used in 
composition. 

II. A multiple is the product of all its prime fac- 
tors. 

III. A co6mmon multiple of numbers has in it all 
the prime factors of each of the given numbers. 

IV. The least common multiple of numbers is the 
product of all the prime factors of each of the numbers, 
each factor being used in the composition the greatest 
number of times it occurs in any one of the numbers. 

V. A factor of a number is a factor of any multiple 
of that number. 

VI. A common factor of numbers isa factor of their 
sum. 

VII. A common factor of two numbers is.a factor 
of their difference. 

VIII. The greatest common factor of numbers is the 
product of all their common prime factors. 


[Exercise in forming composite numbers, common 
multiples, ete. ] 


> LY—INVOLUTION. 


94. Power. A power of a number is the number itself 
or the product of the number by itself one or more 
times. 

95. involution. The process of forming a power is 
called Involution. 


wr pe ~~ ee oS Pa ee 46 
“sed Ler TE Pid Br rge. mm es " 
eu oa OW aoe oY) 
A a" 


54 NUMBER PROCESSES. 


96. Root. One of the equal factors used to form a 
power is called a root. 


97. Second Power. The product of two equal factors is 
cailed the second power, or square of either of them. 
93. Second Root. Either of the two equal factors 
that compose a second power is called the second, or 
square root of the given power. 


99. Higher powers are formed by the composition of 
a greater number of equal factors. Every such power 
ix named by the ordinal of the number of equal factors 
used in the composition of the power. 


100. A root is called the third, fourth, etc., if it be one 
of three, four, etc., equal factors that compose a power. 


101. Any number is called both the: first power and 
the first root of itself. A first power can enter into a 
synthesis with its equal and thus become a component 
of a higher power but, is itself not formed by involu- 
tion. 

102. The index of a power is a small symbol of num- 
ber written atthe right and above a given number and 
indicates that the number is one of as many equal fac- 
tors as there are integral units expressed by the index. 
Thus,.2°==3)-6'==36" etc. 


103. Table of Squares from I? to 24’. 


v= 1 9= 81 17°=289 
= 4: 10°=100 18°=324 
3 9 117=121 19°=361 
4?—16 12'=144 20?=400 
52=25 13*==169 D144 
6'=36 14’=196 22484 
749 15°=225 23%=:529 
8'=64 16°=256 24576 


INVOLUTION, | 55 


104. Table of Cubes from I to 9°. 


j?= 1 LS pome td 
pee, 5 6216 
i aa C048 
4*—64 §=—512 
9729, 
Exercises, 

tae eo what? 16. what? 
eee at hh Poms re 
See Dey A" hin RRO We dem 
Be ee) oS Torrens 8 
ON ae aera a 2,0 BS AF fe po 
SC) tae yD Re Oe Senn ead 
ee ast BP fBa3! ies 
Brae)“ Baa tees Pint 
Be hO2tss 's. ‘ DS Ne geek 
Ohio. { Bop tipte rte 
Piao. Sen witites tt 
Maas“ A a 
ieaea Pes aie We 
oeoittee. Hh Tee 
eo Gt =) << BUR Ae 


105. Remarks on Synthesis. 

1. Numerical synthesis classifies itself under four heads: viz.. 
addition, multiplication, composition and involution. 

There is, however, but a single method of synthesis, and 
that is addition. In multiplication (by aninteger),including com. 
position and involution, a sum is remembered, this sum having 
been previously found by the synthesis called addition, 

2. In addition two numbers are combined by one impulse of 
the mind without regard to the equality of the numbers. 

3. In multiplication (by an integer) a given number of equal 

numbers may be thought as combined at once. the number 
of equal numbers is not limited to tv, but may be any number 
whose sum, found by constant addition, can be given immedi- 
ately from memory. 


4. In composition a definite number of factors are used in 
continued multiplication 

5. In involution a definite number of equal factors are used 
in continued multiplication 


56 NUMBER PROCESSES, 


The Analytic Processes. 
I—SUBTRACTION, 


[Norr. In addition the mind thinks the sum of 
twvu addends or of two parts of two addends immedi- 
ately upon perceiving the two addends or the two 
parts. This process may be repeated until the mind | 
associates not only the two addends with their sum, 
but also associates the three numbers, viz.—the sum 
and its two addends. When this association is perfect, 
the mind is able to recall any one of the three num- 
bers if the other two are known: i. e., either of the two 
addends are as readily recalled when the sum and the 
other addend are given, as is the sum when its two 
addends are known. 

The process or act of recalling from memory one of ‘he two 
addends of a sum when the sum and the other addend are 
known ts called subtraction. 

It is thus seen that in subtraction, as in addition, 
there are involved two addends and their sum. ] 


Formal Definition of Terms. 
106. Difference. The difference between two num- 
bers is the number which added to one of them makes 
a sum equal to the other. 

Remark. The difference between two numbers is sometimes 
defined as the numerical excess of one number over another. 
107. Subtraction. Finding the difference between two 
numbers is called subtraction. 

Remark. Subtraction is sometimes defined as the taking o° 
one number from [out of] another. 

108. Minuend. The swm involved in subtraction is 
called the minuend. 


Remark. The minuend is sometimes defined as the number 
to be diminished. 


109. Subtrahend. The given, or known addend in- 
volved in subtraction is called the subtrahend. 


4 eee a Pes Th, 
ci ss Sh ae ea 


SUBTRACTION. 57 


110. The Mental Act. The mental act of subtraction 
consists in recalling from memory the difference when 
the minuend and subtrahend are known. 


Remark. The act of subtraction as performed by the suffic- 
iently matured mind has its genesisin counting (with objects or 
otherwise) from the given addend (subtrahend) to the given 
Sum (minuend): or by beginning with the minuend and count- 
ing out of it the number of units in the subtrahend, and then 
counting the units of the minuend that remain. 

It is believed that the mind more readily perceives the organic 
relation of subtraction by viewing it as the correlative of addi- 
tion; e. g., 

1. Three blocks and two blocks are how many blocks? 

Ans. Five blocks. 

2. Three blocks and how many blocks are five blocks? 
Ans. Two blocks. 


fll. Reduction. 


1. Since the minuend isasum, it is greater (in arith- 
metic) than the subtrahend. It sometimes occurs, 
however, that there is a less number of units of some 
order in the minuend than of the same order in the 
subtrahend. In such case the minuend must be pre- 
pared before the subtraction can be effected. This 
preparation consists in reducing a unit of the order 
next higher in the minuend to units of the required 
order and combining them with the units of that order. 
If there be no units of the order next higher in the 
minuend, the work of reduction must begin at the first 
order up the scale in which numerical value is thought. 
A unit of that order is reduced to units of the next 
lower ; one of the units resulting from this reduction 
is then reduced to units of the next lower, and so on. 
until the number of units of each order in the min- 


5 


01d SEES Rae i? Bett ie ot 
ie " Meare Se 65 9 set Cue: 
aye St Oa, sek We 


58 NUMBER PROCESSES. 


uend equals or exceeds that of each order in the sub- 
trahend. The subtraction is then readily effected. The 
reduction involved is reduction descending. 


2, The reduction mentioned in 1 may be avoided by 
adding 10 units of the deficient order in the minuend 
to the units of that order, and then compensating this 
aldition by adding 1 unit of the next higher order to 
the subtrahend. © [Prin. IT] 


3. To prepare fractions for subtraction it is neces- 
sary to reduce the minuend and subtrahend to like ~ 
fractional units. If each fraction involved is in its 
lowest terms the reduction preparatory to subtraction 
is reduction descending. 


4. In subtraction of compound numbers the same 
necessity for reduction often exists. It is effected in 
the same manner as indicated in 1. 


12. The Sign. The sign of subtraction is a single 
dash (—) placed after the minuend and before the 
subtrahend in an indicated subtraction. It is read 
Minus. | 


113. Principles. 
I. Only like numbers are used in subtraction. — 


II. If the minuend and subtrahend be equally in- 
creased, the difference between the sums thus obtained 
equals the difference between the minuend and sub- 
trahend. 


III. If either or both minuend and subtrahend be 
used in parts, the partial differences combined equal 
the entire difference. 


SUBTRACTION. 


Exercises. 


‘H4. Subtraction of Decimal Numbers. 


I. IL. 
1. 784— 234 1. 1000— 492 
2. 894— 635 2. 1200— 576 
o. 445— 232 8. 1450— 459 
4, 567— 456 4, 145.7— 82.8 
5. 445— 236 5. 15.62— 9.75 
6. 546— 459 6. 1625— 1000 
t. 821— 379 Tt 17.64— 8.97 
8. 73.5—48.6 8. 2000— 945 
9. 8.24—4,52 9, .211.1— 67:9 
10. 937— 487 10. 2211— 1100 
III. IV. 
1. _13814— 1099 1. 4567— 3.019 
2. 145.6—139.9 2. 45.61— 22.28 
3. 1676— 1554 3. 567.4— 355.6 
Se liis— 929 4, 56.824—22 346 
5. 79845—34545 5. 84562— 77565 
6. 89256—44579 6. 298.74— 48.25 
7. 56733—25876 7. 76.29— 34.75 
8. 45892—34783 8. 87291— 34568 
9. 42.345— 8.898 9. 25371— 77777 
10. 3442— 2009 10, 27623— 78564 
¥.. VI. 
1. 385762— 213858 1. 10000— 8972 
2. 8684,2— 2315.6 2. 100.000— 12,584 
o  4089— 3478 3. 18887— 12,384 
4. 96.788— 4,239 4, 21472— 10.019 
0. (621— 5487 5. 367.82— 18.996 
6. 8123— 4565 6. 1892— . 847 
7. 56782— 34567 7. 476792—341.453 
8. 588721— 34562 8. 100841— 99872 
9. $239.46—321.789 9. 10084— 9721 
10. 721544— 362847 10. 56321— 24525 . 


60 


NUMBER PROCESSES. 


Subtraction of Fractions. 


115. 


Wakee cao ho te CYOD tt 


ool 


o> rfF ech clea 0 cola 


Kasonad 
SA spt 


+ 4 
i 


ewe FIM eho nl we aH 


frst het 


SHO och wefs0 fA ofS che 


AN od WH id 


ls 


Ae 


' 
|" 


wefro miko CNY echo who 


ca =) ae 
lo on ” Ile ole: 
—Oaona 
en | ro at 


ake aes raf? rit ei) oI 


Palen cae 


mho = 
lor cds! to i i 


rac +15 


rie che Hes isle ee cc * mica ert al cows cys he 


Bissau plese 


wl CO ste acho COD ~<H sht- 190 ala DP ho ~xH 


EVs, 


Haost io SK DOS Ona 
kane Ga 


hs nilot inks oho Hes miko HH NT SH etl CY) aha 


| ; : 


who OD BO ela cho {~~ cf a tho ho ho sO 


Tit. 


ANAMDHTIDOOMAGSOnN 
2 ce oo em 


SUBTRACTION, 


116. Subtraction of Compound Numbers. 


(1) 
gal. qt. pt. 
nrc bes <Q 
emer), 1 
(3) 
ip. 02. pwt. : ‘gr. 
Be rd 16%: 13 
apie tet oly. 3° 12 
(©) 
bu. pk. qt 
Deets. yt 
7 es als 
(7) 
bu. pk. qt. 
Ba Dx O 
areas): ! 2 
(9) 
Foil tur. 1d: 
hy Mas ga CY 
Geeross! 15 
(11) 
cu.yd. cu.ft. cu.in. 
14 10 50 
10 16 63 
(13) 
sq.yd. sq.ft. sq.in. 
8 fi 19 
4 6 25 


(2) 
pk. 
2 


bu. 
4 
1 3 


(6) 
OZ. 


ee 
6-8 


qt. 
7 
6 


min. 


OU 
15 


16 
18 


pwt. 


61 


sec. 
20 
40 


> el oe 2 ee ee 
4 Blaster Sats hy ie 
f ‘ 


62 NUMBER PROCESSES, 
(15) : (16) 
mis stur, oes yd. fhe 
2 7 30 5 1 10 
1 6 25 2 2 wel 
(17) (18) 
Ib. 0z. ‘pwt. gr. cu. yd. « CU tiuGiesy 
cna AO Peete AA 2 10 75 
i Rade maes Curae By Ci vp he! 1 18 150 
(19) (20) 
Ibi ze ter bu. pk. qt. 
TaD rae Ae 4 1 7 
PAPE sve ENON Al Rs 3 2 3 
(21) (22) 
Vol. ay Sete wn Ib 3 3) 
8 1 10 Tee 5 1 
6 2 11 Due 6 2 
(23) (24) 
gal. qt. pt... gi lb." 02) pet pam. 
ere apa ie gee 8s 7 he 
hia pany Ike 65.8) Sabor 
(25) (26) 
palodty pirreeh yds fpawee 
2 eles earerie a a eee 
pei 8 We ANE a: Me ty Meee: 
(27) (28) 
eam mama ed SE mi. fur. rd 
farts ee ai ea Docs eee 
BG oo Uae 4. Seek 


DIVISION. 63 


II—DIVISION. 


117. The Genesis and Nature of Division. 


1. If a given subtrahend be taken from a given 
minuend, and again be taken from the remainder, and 
again be taken from the second remainder, and so on 
until the given minuend is exhausted or gives a remain- 
der less than the constant subtrahend, the several sub- 
tractions viewed together are called a constant sub- 
traction. 

If, when the minuend and constant subtrahend are 
known, the mind gives from memory the number of 
subtractions necessary to exhaust the given minuend ; 
or, if the minuend and the number of subtractions that 
can be made are known, and the mind gives from 
memory the constant subtrahend, the act is called 
division. 

os [The primary notion of division is, without doubt, 
that of separating a number into equal parts. The 
word itself makes this evident. This separation may 
be made for either of two purposes; viz., to ascertain 
how many such parts there are in the number, or to 
find how many [units] there are in one of the parts. 
The problem of division may be solved by subtraction ; 
but the process which we call division is not based, 
primarily, upon subtraction but upon multiplication. 
—O ney. ] 

8. In multiplication the mind thinks the product 
of two factors immediately upon knowing the two 
factors. (This step may be taken in parts. Prin. VI, 
page 48.) This process may be repeated until the 
mind so associates the three numbers, viz. :—the two 
factors and their product, that any one of them may 
be at once recalled when the other two are known. 


64 NUMBER PROCESSES. 


The act of recalling from memory either of the two factors 
of agiven product when the other factor is known, is 
called division. 

The given product involved in division is called 
the divilend, the known factor the divisor, and the 
required factor the quotient. 

4. In its relation to multiplication and in the usage 
of mathematicians, division has a broader meaning 


than the separation of a number into parts. Independ- _ 


ently of concrete application, the scope of division, 
as a number process, is found in viewing the quotient 
as the ratio of the dividend to the divisor. [The 
divisor is also the ratio of the dividend to the quotient, 
and hence is the reczprocal! of the ratio of the quotient 
to the dividend. ] 

The quotient, therefore, sustains the same relation 
to the dividend that one does to the divisor; and divis- 
ion may be defined as the act of obtaining from the 
dividend, a number that is indicated by the reciprocal 
of the divisor. 

Illustrations. a. If the divisor be 2, the quotient is 4 
of the dividend. 7 

b. If the divisor be ?, the quotient is 4 of [times] 
the dividend. 

c If the divisor be 4, the quotient is 3 times the 
the dividend; etc, — 


Formal Definition of Terms. 

118. Quotient. The quotient of one number by another 
is a number that sustains the same relation to the first 
number that 1 does to the second. 

Remarks. 1. The quotient of one number by another is the fac- 
tor which, used with the second number, will produce the first. 

2. The reciprocal of a number is the factor by which if the 
given number be multiplied the product is 1. 


DIVISION. 65 


HD. Division. Finding a quotient is called division. 


Remark. In the light of remark under Quotient, division may 
be defined as finding one of two factors of a given product when 
the other factor is known. 


120. Dividend. The number to be divided is called 
the dividend. 


Remark. The dividend is the given product of which the 
divisor is the known factor. 
121. Divisor. The term divisor is usually defined as 
the number by which the dividend is divided. 


Remark, The divisor is the factor given with the dividend 
to determine the quotient. 


122. The Divisor of Units’ Order. If the divisor be 
units of the first, or units’ order, the quotient is of the 
same order of units as the dividend. 

Illustrations. 12 u+38—4u. 12 tens+3—4 tens. 

12 fifths+-3—4 fifths 12 bu.+3=—4 bu. ete. 


Remark. The division of 2 fifths by 3 is usually effected by 
multiplying the denominator by 3, thus obtaining 2 jifteenths for 
the quotient. If, however, the given dividend be first reduced 
to fifteenths, the division may be effected as in the preceding 
illustrations. 


123. The Divisor of Higher Order Units. If the divisor 
be units of an order higher than the first or units’ 
order, the quotient is of an order of units correspond- 
ingly lower than those of the dividend. 
Example. a. 12 tens + 3 tens = what? 

. Solution. 12 tens + 3 = 4 tens, but since the divisor 
is of tens’ order, the quotient is of the order next 
lower in the decimal scale than the dividend, and is, 
therefore, 4 wnits. 

[Or the quotient is of an order of units 7; as great 
as tens, and is, therefore, 4 wnits. ] 


66 NUMBER PROCESSES. 


Example. 6. 12 hundreds + 3 tens = what? 
Solutton. 12 h +3 = 4h, but since the divisor is 
of tens’ order, the quotient is of the order next lower 


in the decimal scale than the dividend, and is, there- 


fore, 4 tens. 
| Example. c. 12 units + 3 tens = what? 
— Solution. 12 w+3=—4 wu, but since the divisor is 
of tens’ order, the quotient is of the order next lower 
in the decimal scale than the dividend, and is, there. 
fore, 4 tenths. 

Example. d. 12 fifths + 3 tens = what? 


Solution. 12 fifths +-38=4 Jifths, but since the divisor 


is of tens’ order, the quotient is of an order of units 10 
as great as fifths, and is, therefore, 4 fiftieths. 


124. The Divisor of Lower Order Units. If thedivisor 
be units of an order lower than the first or units’ order, 
the quotient is of an order of units correspondingly 
higher than those of the dividend. 

Example. a. 6 units + .8 = what? 

Solution. 6u + 3 = 2 u, but since the divisor is of 
ede order, the quotient is 2 tens instead of 2 units. 

*, 6 + 3 = 2 tens, or 20. 

Example. 6. 6 wnits + # = what? 

Solution 6u+8=2 u, but since the divisor is 
of fourths’ order, the quotient is 2 fours instead of 2 
ones. ..6 + 2 = 2 fours, or &. 

Example. c. $+ $= what? 

Solution. §$ + 3 = 2 sevenths, but since the divisor 
is of fourths’ order, the quotient is of an order of units 
four times as great as those of the dividend, and is, 
Cette 2 four-scvenths, instead of 2 (one) seventh. 
ee four-sondriths: or 4. 


DIVISION. | 67 


Example. d. % + # = what? 


Remark. If the numerator of the dividend is not exactly 
divisible by the numerator of the divisor, the dividend may be 
reduced to an order in which such division can occur. 3 fifths 
=6 lenths, 6 tenths may now be divided by 2 sevenths by the 
form of solution given for example c. 

Examples like c and d are usually solved by multiplying the 
dividend by the reciprocal of the divisor or by some of the 
solutions suggested under Division of Fractions in this book. 


125. Principles. 


I. The quotient sustains the same relation to the 
dividend that 1 does to the divisor. 

Il. If the dividend be divided in parts the sum of 
the several partial quotients obtained is the entire 
quotient. 

III. If the dividend be divided by the factors of the 
divisor used in continued division, the final quotient 
is the quotient of the dividend by the entire divisor. 

Remar’. In applying this principle a remainder may occur 
upon dividing by one or more of the factors of the divisor. 
These partial remainders do not constitute the ultimate or true 
remainder. [See Appendix. ] 

IV. A fraction is divided by an integer by dividing 
its numerator, or multiplying its denominator by the 
integer. 

V. A number is divided by a fraction by multiply- 
ing the number by the reciprocal of the divisor. 


126, General Principles of Division. 


Remark, The following six principles are called general prin. 
ciples. 

I. Multiplying the dividend by any number multi- 
plies the quotient by that number. 


68 


II. Multiplying the divisor by any number divides 
the quotient by that number. 


LE 
number does not change the quotient. 


IV. Dividing the dividend by any number divides 
the quotient by that number. 


V. Dividing the divisor by any number multiplies 
the quotient by that number. 


VI. Dividing dividend and divisor by the same 


NUMBER 


PROCESSES. 


Multiplying dividend and divisor by the same 


number does not change the quotient 


127. 


Division of Decimal Numbers. 


10. 
A bie 


pm 
Ee 


09 CONTR OU O99 DO 


Oo Oe re 


Ti 
9642-~ 6 


8379+ 7. 


3246 8 
jee he se) 
122436--12 
4368-13 
3690--15 
8024-17 
37620411 
108198--18 
3224214 
8171216 


LIT: 


41625-~-37 
3971038 

4879-39 
67322 —41 
57276+43 
4936846 
60904--46 


7670447 


Exercises. 


NTS SERS seg el 


9 Co sT OSU Go bo 


IT. 


3168-24 
5575-25 
6396--26 
6777-27 
10136--28 


11097749 


19778+31 
7686432 
10725--33 
20808--34 

7875-35 
20692--26 


IV. 


811332372 — 
301911641 
259374622 
126294582 
273176463 
- 9384949-+307 


7287-+347 
2555 +365 


DIVISION. 


V. VI. 
ees LO20=-537 EO ggite dt & bro, WARS. oy, 
22. 397 10-738 2. 301911~644 
3 4879-839 BS POUS T4267 
4 567322---241 Ar A DB2O 4 895 
5. 957276+543 Mee ol TOs-Oot 
6. 849368-—846 6. 9334949—370 
7. 360904146 1201417287473 
8. 276704447 8. 5942555658 
B. , 4280-288 9. 62377495 
BU 2345005 70 10. 25900625 
Bhs 927 kl 83 1 117.128 488"957 
iT Opps 20 12. - 345676356 
tio oo4oy 779 13. 8866709--907 
14. 798537464 14. 9786567~—755 
15. Pep OD 1b. = 36784 .08 
Po pend ep .20 16. Se Ole hG 
Pye Dsos-. 125 17. 12942. 009 
18.°».172:8-- .12 18. GASES ONS 


128. Division of Fractions. 


oh CO colo 


1 II rer 
1 2+3 1. 9+% 1 ¢+ 
2. $2 2. 10+4% 2. $-- 
3. $+ 3. 12+4 o. $F 
4, $20 4, 15+2 4, %-- 
5. 474 5. 16-4 5. 2+ 
6. 13-5 6. 18+4$ 6. 5+ 
7. £+4 7. 12-+3 7. 33+ 
8. €+3 8. 8+4 8. 35+ 
9. 3+3 9. 10+ 9. 4 
10. 49--5 10. 9+4 10. 4+ 
11. 4+4 11. 16+} 11 3+ 
12. 38-2 12. 20++4 12. .9+ 


Tyee AEP orks cako Hy KO 


sie AR 


70 


NUAIBER PROCESSES. 
IV; V VIL 
1. 32+4 1 4+2 "1. 3+3 
2. Y=-4 2. $+4 2. .6+ 2 
3. $+ 2 3. #72 3. 4+4 
4. {+32 4. 35+ys 4. 3+3 
5b. 2+4 Oo PF 5. +35 
6. +1 6. 3--: 6. .4+235 
8. s5+.-2 8. #+4 8. 3+; 
9. 4.4 9. $+) 9. $+ 35 
10. §+3 10. 44+} 10. .7+ 3 
ll. 4+.3 ll. $+8 ll. 2+? 
12. 6+ 2 12. +4 12. 4+ 3 
Division of Compound Numbers. 
bu. pirat. 
1, Divide 5 3 2 Wye: 
gal; 2 qt. pt. 
2. Divide 8 3 2 by 4. 
Ib.. oz. pwt. gr. 
or) Divide 28:23 210s 1s ae 
NOs area ais ih 
4. Divide 94.25 922020. Gen aaanes 
ye te Tt am 
5 Divide 10 2 3 by 9; 
gal. gt. pt. gi. 
6: Divide “os Flue Ws fea byes 
lb. oz. pwt. gr. 
fe Divide 18 20)8)--06 12 yee 
. sis RBs 8 Bip ge 
8..)° Divide. [405,475.10 pyie 0 
bus peau gh 
9. > Divide yctis ss pol Dye 
14.39 Waeets SV a wie TS v0 D 
10. Divide 1 Does 8 4 DYES 


Divide 
Divide 
Divide 
Divide 
Divide 
Divide 


Divide 


Divide. 


Divide 
Divide 
Divide 
Divide 

Divide 
Divide 
Divide 


Divide 


DIVISION. 
hr. min. sec. 
Livtcosn Ose DY. 9. 


bi pk qt. 
8 


gal. qt. pt. 
ae Retr AU 
Ib. Oz: pit. er. 
YA PS Se ee 


Maoh Eta 9 im: 
9 il Ae by 6, 


Daly DK Ot, 


13 3 Bil Dyas de 
Tin shuns we ras 

17 3 5 LES 8 ga 
LAD SUF aii The eds Dede A 


Lee tic Sunred by. 9: 


VOb 0S). Div.byy) ST. 


SEF Una se: LO: DY. 2s 
ele Dia UL 

15 3 bby -8, 
gal. qt. pt. gi. 

eee oh Mok: Glee DY ik On 
sq. yd. sq.ft. sq.in. 

121 2 Gus bye LO; 
bute pk. qt. 

ome moumeh a! Va. -O 
da. hr. min. 

a 20 by 15 
oz. pwt. gr. 

6 3 8 by 4 


71 


72 NUMBER PROCESSES. 


III—DISPOSITION. 


130. Definition. The separation of a composite num- 
ber into its factors is called disposition, or factoring. 

Remarks. 1. Disposition is a phase of division viewed as the 
process of finding the factors that compose a multiple. 


2. Disposition is the reverse of composition. In composition 
the factors are given to find their product, while in disposition 
the product is given to find its factors. 


3. In disposition the factors are found by dividing the given 
multiple by any exact divisor of it. The quotient thus found 
is divided by any exact divisor of itself, etc., until the required 
factors are found. The several divisors used and the final quo- 
tient are the factors of the given multiple. 


If the prime factors are required, the several divisors used and 
the final quotient must be prime numbers. 


4, A numberis said to be divisible by another if the quotient 
is an integer. 


This is a limited meaning of the word divisible. In the light 
of the definitions and principles of division, any number is 
divisible by any other number. 


131. Principles. 


I. A number that is divisible by two or more num- 
bers in continued division, is divisible by their pro- 
duct. 


II. A common divisor of two numbers is a divisor 
of their difference. 


III. If a number be divided by one of its prime fac- 
tors or by the product of two or more of them, the quo- 
tient is the remaining prime factor or the product of 
the remaining prime factors of the number. 


IV. If the product of two factors be divided by either 
of them the quotient is the other. 


DISPOSITION. | 73 


V. If the product of more than two factors be divided 
by one of them, the quotient is the product of the other 
factors of the number. 


132. Divisibility of Numbers. 


Remarks. 1. There is no general method devised whereby 
the factors of a multiple may be readily found ; nor is there any 
means whereby a number is known to be composite. Certain 
numbers, however, possess characteristic marks denoting that 
they are composite. A few of these will be discussed. 

2. An integer whose units’ figure is 0, 2, 4, 6 or 8 is called an 
even number. All other numbers are called odd numbers. 


I. Aneven number is divisible by 2. 
Explanation. Any integer whose units’ figure is 
0 may be thought as a number of tens and hence is a 
multiple of 2. Every even number is, therefore, a 
number of tens plus 2, 4, 6, or 8, and hence the sum 
(an even number) is divisible by 2. 


II. If the sum represented by the digits of a num- 
ber be divisible by 3, the number is a multiple of 3. 


Explanation . 
( 40004 x 1000=4 (999-++1)=4« 999-+-4. 
ae 500=5x 100=5 ( 99-+1)=5x 99-5. 
Meee) 10—-1/( 9--1)=1X - 9-+1. 
Por 2. 

Upon separating any number, as 4512, into parts as 
indicated above it is observed that the last member of 
each of the continued equations is separated into two 
addends. The first of each of these parts is seen to be 
a multiple of 3. The other parts, together with the 
second member of the last equation, are represented by 
the several digits of the given number. If, therefore, 
the sum represented by these digits be divisible by 
3, the given number is divisible by 3. 


ree Sitar Wh te pS . 
Monee ei the. iris MS te 
Noe Soe aS 


74. NUMBER PROCESSES. 


A Second Explanation. The place value of any figure 
in the decimal representative scale is a multiple of 9 
and hence a multiple of 8. Now if the sum of the 
form values of the figures representing a decimal num- 
ber be a multiple of 3, the sum of those values and the 
place values represented is a multiple of 3. } 


III. A number is divisible by 4 if its two right hand 
figures are zeros or represent a multiple of 4. Why? 

IV. A number is divisible by 5 if its units’ figure 
is 0 or5.. Why? 

V. A number is divisible by 6if it be even and a 
multiple of 8. Why? 

VI. A number is divisible by 7 if once its units + 3 
times its tens + 2 times its hundreds + 6 times its 
thousands + 4 times its ten-thousands + 5 times its hun- 
dred-thousands + the numbers represented by the suc- 
ceeding figures multiplied, respectively, by the series 
of multiphers named above, be a multiple of 7. 

VII. A number is divisible by 8 if its units’, tens’ 
and hundreds’ figures are zeros or represent a multiple 
of 8. Why? 

VIII. A number is divisible by 9 if the sum of the 
numbers represented by its digits bea multiple of 9. 

Remark, This may be explained in a manner similar to that 
given for divisibility by 3. 

IX. A number is divisible by 10 if its units’ Sgurs 
is 0. Why? 

X. A number is divisible by 11 if the difference 
between the sum of the numbers represen by the 
digits in the odd places and the stim of the numbers 
represented by the digits in the even places is nothing 
or a multiple of 11. 


_ EVOLUTION, 75 
XI. A number is divisible by 12 if it be a multiple 
of 83and 4. Why? 


133. General Remarks. 


1. A number is prime if it fail of division upon being tested by 
every prime number up toa divisor that gives a quotient less 
than thedivisor. Why? 

2. In factoring a number the pupil should always test it by 
the conditions herein given, and not guess at its factors until the 
tests, as far as known, have been applied. . 

3. Pupils should learn the prime factors of every composite 
number from 4 to 100. 

[ Exercises. ] 


IV—EVOLUTION. 


124. Definition. The separation of a power into the 
~ equal factors which compose it is called evolution. 

(1.) Each of the equal factors found by evolution 
is called a root of the power from which it is evolved. 

(2.) A root is called the second, third, fourth, etc., 
according as it is one of two, three, four, etc., equal 
factors that compose a power. 

Any number is called the first root of itself. 

(3.) The index of a root is a fractional unit writ- 
ten at the right and a little above a written power and 
indicates by its denomination the root required. 

(4.) The radical or root sign (7/) is often used to 
indicate a root. If used alone before a number it indi- 
cates the second, or square root. If a root other than 
the second is required, the radical sign has placed above 
it the denominator of the fractional unit which denotes 
the required root. 8*=)/8=2. 

(5.) Exponent. The index of a power or of a root 
is often called an exponent. 


76 NUMBER PROCESSES. 


(6.) If a number be affected by a fractional expo- 
nent other than a fractional unit, the ordinal of the 
numerator of the exponent is the power to which the 
number is to be involved, while the ordinal of the 
denominator is the root to be evolved from that power; 


or, the ordinal of the denominator of the exponent is. 


the root to be evolved from the given number, while 
the ordinal of the numerator is the power to which 
that root is to be involved. Thus, 8“ indicates that 
the third root of the second power of 8 is required ; or 
it indicates that the second power of the third root of 
8 is required. In either case the result is 4. 


(7.) In algebra numbers having negative expo- 
nents are treated. 

A negative exponent indicates the power or root, 
as the case may be, of the reciprocal of the number 
affected; e. g., 4>? indicates the second power of 4 
while 8“ indicates the third root of 4. 


[ Exercises. ] 


135. Remarks on Analysis. 


1. The analysis of numbers classifies itself under four phases, 
Viz.: SUBTRACTION, DIVISION, DISPOSITION and EVOLUTION. 

2. In subtraction a number is thought as separated into two 
parts without regard to the relative value of the parts. 

3. In division a number is thought as separated into a defi- 
nite number of equal parts. ! 

4. In disposition the factors of a given composite number are 
found by continued division. 

5. In evolution one of a definite number of equal factors which 
compose a power is found. 

6. Any root is readily found by reversing the steps taken in 
forming the corresponding power. 


ey ea les 


DIVISORS AND MULTIPLES, 


Greatest Common Divisor. 


136. Definition. The greatest common divisor of given 
numbers is the greatest number that is contained an 
integral number of times in each of them. 


137. Principles. 

I. The product of all the common prime factors of 
given numbers is their greatest common divisor. 

II. A divisor of a number divides any multiple 
of it. 

III. A common divisor of two numbers divides their 
difference. 
IV. A common divisor of given numbers divides 
their sum. 


78 DIVISORS AND MULTIPLES. 


138. Methods of Finding g. c. d. 


(1.) In the light of Principle I, we find the prime 
factors of the given numbers and take the product of 
those factors that are common to allthe numbers. This 
product is the greatest common divisor of the num- 
bers. 


(2.) The common factors of given numbers may be 
found by dividing them by a number that is seen to be 
a common factor of them. Divide the resulting quo- 
tients in the same manner, and so continue until quo- 
tients are obtained that are relatively prime. 

' The divisors used are the common factors sought, 
and their product is the required greatest common 
divisor. [Prin. I.] 


Example. Find the g. c. d. of 24, 30 and 42. 
Blackboard form. 
2)24.... 80.5.5 42. 
S12 bee wae 
roe ge Rea ee 
“24 B= Oeap. CAL 
(3.) The.“ division” method is effected by divid- 
ing the greater of two numbers by the less (comparin: 
but two of the numbers at a time); and then dividing 
the divisor by the remainder, and so continuing to 


divide the last divisor by the last remainder until there. 


isno remainder. The last divisor is the greatest com- 
mon divisor of the two numbers compared. This divisor 
is then compared with another of the given numbers 
(if there be more than two) by division as above indi- 
cated, and so on until the numbers in any given case 
are all used. The last divisor thus found is the g.c.d. 
of the several given numbers. 


SO ea : 
re ri. “ 


GREATEST COMMON DIVISOR. 79 


Example. Find the g. c. d. of 260 and 716. 
Blackboard form. Thought form. 
260| 21716. Since 260 is the greatest divisor of itself, 
ne 520, if it divide 716, then is 260 the g. c. d. 
ae 196 of itself and 716. Upon trial we find 
te that 260 does not divide 716, but that the 
oer multiple of 260 in 716is520, By 
Prin. Il we know that the g. c. d. will 
divide 520, and by Prin. III we know that it will 
divide 196, the difference between 716 and 520. 196 
is the greatest divisor of itself. Now if it will divide 
260, it will also divide 2 times 260--196, or 716, and : 
ence will be the required g. c. d. 

Upon trial we find that 196 does not divide 260, but 
by Prin. III we know that the g. c. d. will divide the 
difference between 260 and 196, which is 64. 64 isthe 
ge. d. of itself. Now if it will divide 196 it will divide 
the sum of itself and 196, or 260, [Prin. [V.] and also 
2 times 260+196, or 716, and hence will be the required 
g.c.d. 64 does not divide 196, but the greatest mul- 


tiple of 64 in 196 is 192. By Prin. II we know that the 
ge. c. d. will divide 192, and by Prin. III we know that 
it will divide the difference between 196 and 192 which 
is4. 4 is the greatest divisor of itself. Nowif 4 divide 
64 it will divide 3 times.64, or 192, and also the sum 
of 4and 192 which is 196, ‘and also 196-+64, or 260; 
and also 2 times 260-+-196, or 716, and hence is the 
required g.c.d, 4 divides 64, hence 4 is the g. c.d. of 
260 and 716. 

Remark. The work may often be shortened by taking the 
multiple of the divisor that is nearest the value of the dividend, 
and finding the difference between that multiple and the divi- 
dend for a new divisor. 

In the above example the multiple of 260 that is nearest the 
value of 716 is 780, and the difference between 780 and.716 is 
64. We thus see that the g.c. d. of the given numbers may be 
found by two divisions instead of three as in the solution 
given. 


80 DIVISORS AND MULTIPLES. 


[ Exercises. ] 


Find the g. c. d. of the following: 


1.0.15, Bt) 984 9. 576 and 168. 
2. 24,16 12, 20. 10. 121 “ 495. 
853.63: 27) 86: 11.21 ee 
4. 20, 42, 54 12. 260 “ 416 
5. 15, 40, 55, 60, 75 13. 125 “ 500 
6. 64, 72, 100, 96. 14. 294 “ 472. 
7. 75, 135, 400. 15. 108 “ 146. 
8 72, 84, 132 16. 1245 “ 600. 


Least Common Multiple. 


139. Definition. The least common multiple of given 
numbers is the least product of which each of them is 
a factor. 

140. Principle. The least common multiple of num- 
bers is the product of all the prime factors of each of 
the numbers, each factor being used the greatest num- 
ber of times that it occurs in any of the numbers. 


Example. Find the l. c. m. of 12, 15 and 25. 


Blackboard form. Thought form. 
12—=22x3 Since the required Lc, 
15=3 Xo. m. contains 12, it con- 
25=0 XO. 


tains the prime factors 
, of 12 which are 2, 2and 
3. These we take as factors of the l.c.m. Since the 
l. c. m. contains 15, it contains the prime factors of 15 
which are 3 and 5. We have 3 asa factor of the lc. 
m. so we take 5 as one of its factors, and thus have the 
prime factors of 15 as factors of the 1 c. m. Since the 
1 c. m. contains 25, it contains the prime factors of 23 
which are5 and 5 We have one 5 asa factor of the 
l. c. m., so we take another 5 as one of its factors and 


IX2K3X5 X5—=300=L. ¢. m| 


LEAST COMMON MULTIPLE. 81 


thus have the prime factors of 25 as factors of the l.c m. 
We now have as factors of the required l. c. m. all the 
prime factors of 12,15 and 25 and no other factor. The 
product of these factors=300, the required 1. c. m. 


Remark. If numbers are not readily factored, their l.c. m 
may be found by either of the following methods. The first 
comparison instituted in each method is between two of the 
given numbers Next between the l.c m. of the two numbers 
already compared and the third number. Next between the 
l.c. m. of the first three numbers and the fourth, and so on 
until all the numbers in any given case are used. 


(1.) If one of the two numbers be divided by their 
g.c. d. the quotient contains those factors of the num- 
ber divided that are not found in the other number. 
Now if the undivided number be multiplied by this 
quotient, the product contains all the prime factors of 
_ the two numbers and no other factor, and hence is 
their |. c. m. 

A rule for this method may be formulated thus: 

Divide one of two numbers by their g.c.d. and multiply 
the other number by the quotient. The product isthe l.c.m. 
of the two given numbers. 


(2.) Since the 1. c. m. of two numbers contains all 
the factors of one of the numbers and such factors of 
the other as are not found in the first, if the two num- 
bers be multiplied together, the common multiple 
obtained is greater than their 1.c. m by a factor equal 
to their g. c. d. 

Hence the product of two numbers, divided by their g. ¢, 
d. equals their l. c. m. 


[Exercises]. 
Find the l.c. m. 
a2. 5,14; '18. 5. 24,32, 42, 50. 
2. 21,16, 36 6. 15, 28, 40, 64. 
& 18, 36, 44 Ten O; UO LENE 4. 
4. 12, 28, 54 8. 9, 14, 15, 16, 20. 


CHAPTER VIL 
FRACTIONS. 


i41. A Fractional Unit. Theidea one which arises upon 
viewing one of the equal parts into which a whole may 
be thought as separated, is called a fractional unit. 


142. AFraction. (1.) A fraction is a fractional unit 
or a number of like fractional units thought together. 


(2.) A fraction is one or more of the equal, parts 
of a unit. 


Remarte: 1. The primary idea of a fraction is composed of the 
tollowing elements, viz.: 

a ‘tthe idea of a whole divided 

b. The idea of equality of the parts. 

c. Anumber of the parts. 

From this analysis of the primary idea it is apparent that the 
value of a fraction cannot exceed that of the unit Whence the 
fraction is derived. . 

The second definition is based upon the primary idea as thus 
anaiyzed. 

2. Each of many like units may be thought as separated into 
the same number of equal parts and any number of these parts 
may be viewed together; hence the first definition. 

3. ‘The expression 4 iy not to be interpreted as representing 


ten fourths of 1, for one object can be thought into but four 


fourths +4) is to be viewed as expressing a synthesis of ten 


fracticnal one each of which is one-fourth of 1. 


143.- The Notation of a Fraction. [See page 27. ] 


ga) 
G2 


CLASSES, 


‘Glasses of Fractions. . 
144. Decimal and Common. 


1. Based on the decimivl or non-decimil division of 
the unit one, fractions are classified as decimal and 
common. | } . 


2: Dec AL. A fraction whose fractional unit is a 
decimal part of the unit one, is called a decimal frac- 
tion. 


Remarks. 1. A decimal fraction is usually expressed by the 
decimal notation, though it may be written in the fractional 
- form. : . 

-2. A fraction expressed partly; in the decimal and partiy in 
the fractional notation is ealied a complex decimal. _Examples, 
333 0348. 


3. Common. A fraction whose fractional unit is 


other than a decimal part of the unit one, is called a 
common fraction. 


145. Proper and Improper. 


1. With 1 as a basis, or standard of comparison, 
fractions are classified as proper and improper. 


2 Proper A fraction whose value is less than 
1 is called a proper fraction. 

3d. Improper. A fraction whose value equals or 
exceeds | is called an improper fraction. 


Remarks. 1. The primary idea of a fraction is a number of 
the equal parts of a unit. 

' The classification of fractions as proper and improper is thus 
seen to be inconsistent with the primary idea of a fraction. 

2. The definition of a fraction as @ number of like fractional 
units is based upon a secondary idea, viz.. A number of like 
parts of any number of like units. Under this definition any 
ne nber of like fractional units is a fraction. 

The two definitions of a fraction that we have given [ Art. 
12) are fairly r:presentative of the definitions found in the 
text books on arithmetic. Under neither of these definitions 
is there sro 1nd for classifying fractions as proper and improper. 


84 FRACTIONS. 


146. Simple, Compound and Complex. 
1. On the basis of form, fractions are classified as 
simple, conpound and complex. 


2. Srmmp.e. A simple fraction is defined asa frac- 
tion each of whose terms is a single integer. 


3. Compounp A compound eee is defined 
as a fraction of a fraction. 


Remarks. 1. The simplification of a so-called compound fraction 
is effected in the Jight of principies which govern the multipli- 
cation of one fraction by another. 4} of 4 does not express.a. 
class of fractions, but simply indicates that # is to be multiplied 
by # [See Remark 2, under Art. 744] 

2. Some have found ground for this class of fractions in 
thinking a compound fraction as a part of a part instead ofa 
part of a whole. But 3 of # indicatesa part of a part of a 
whole;— i. e., it indicates 3 of # of 1, and hence is as trulya 
part of a whole as is # alone. 


4. CompLex. A complex radtich is defined as 
a fraction having a fraction in one or both of its 


terms. 
Remarks. 1. A complex fraction is read as an expression of ~ 


S is read 2} - 33 and ‘i is read }-- 3. 


In the light of the definition of Pree the first of the 
above so-called complex fractions is derived from the division 
of a unit into 33 equal parts. The mind sees at once the impos- 
sibility of such a division. An examination of the second so- 
called complex fraction given, renders still more manifest the 
absurdity of calling these numerical values fractions. 

2. Since a simple fraction is expressed by a form in distinction 
from comp und and complex, it disappears upon the disappear- 
ance of these as classes of fractions. 

»3. If it is found convenient to use the terms proper, improper, 
simple, compound and complex to distinguish certain phases of 
fractional notation or indicated processes, it may be well to 
retain those terms; but they are certainly not necessary toname 
any of the essential thought elements of a fraction, 


division: e. g. sf 


PRINCIPLES. 8 5 


General Principles. 

Remark. A fraction may be considered as a case of division; 
the numerator being the dividend, the denominator the divisor 
and the fraction itself the quotient. The general principles of 
division become, therefore, the general principles of fractions by 
a change of terminology. 

147. Principles of Multiplication. 

I. If the numerator be multiplied the fraction is 
multiplied by the same number. 

II. If the denominator be divided the fraction is 
multiplied by the same number. 

148. Principles of Division. 

III. If the numerator be divided the fraction is 
divided by the same number. 

IV. If the denominator be multiplied the fraction 
is divided by the same number. 

149. Principles of Reduction. 

V. If both terms of.a fraction be multiplied by the 
same number greater than 1, the fraction is reduced to 
smaller fractional units. 

VI. If both terms of a fraction be divided by the 
same number greater than 1, the fraction is reduced to 
larger fractional units. 

150. Principles of Relation. 

VII. Fractions haying a common denominator are 

to each other as their numerators. 


Remark. This principle may be thus stated: Like parts of 
numbers are to each other as the numbers themselves; e. g., 


the relation of 3 to is the same as that of 2 to 4; i. e., 4 of 2 
bears the same relation to 4 of 4 that the whole of 2 bearstothe 
whole of 4 

VIII. Two fractions having a common numerator 
are to each other as their denominators taken inversely. 


tae Sa A ee (fs ae 


. 


86 FRACTIONS: 


Reduction of Fractions. 
REDUCTION DESCENDING. 


150. Case I. An ee or mixed number to a frac- 
tion. 
Example. Reduce 3 to 8ths. 
Thought form. 

The unit of the number to be reduced is the unit 1. 
The unit of the number to be obtained is 4, a smaller 
unit than the unit 1; hence reduction descending is 
required. Thinking 3 in the form of a fraction by 
giving it 1 for a denominator and multiplying both 
terms by 8 [Prin. V] we obtain 24 as the required num- 


ber. 
Blackboard form. 


51. Other Forms. 
Example. Reduce 38 to 8ths. 


; ea 
*; 1=8 times 4, 
b Pee «¢ 3 24. 
: a a Sins 
ee hee (a 3° 
I= 
f | 3-24 
Remarks onc. 1. Make 1 = $, and then ae a mems« 


bers of the equation by 3. We thus find that 3 =24.. 
2. In any similar example make 1 the first Be es of the 
first equation ; and for the second member take the equivalent 


of 1 in fractional units of the required denomination, © Next 


. ee =i . 
ee 


ee DT a! 
Fg: va» wu “> 


s 


REDUCTION, SF 


- multiply both members of the equation by the integer to be 


reduced. 

3. In reducing a mixed number to a fraction, the fh tae is 
reduced to the denomination of the fractional part by any of 
the above methods and the fractional part is then added. 

152, Casell. <A fraction to smalier fractional units. 
Example. 1. Reduce 4 to 12ths. 
First thought form. 

The unit of the number to be reduced is 4. The 
unit of the number to be obtained is 4, a smaller 
unit than 4; hence reduction descending is required. 
Multiplying both terms of 3 by 4 [Prin. V] we obtain 
=; as the required number. 

Second thought form. 

In the light of Principle V, 4 may be reduced to 12ths 
by multiplying both its terms by such a number as 
will produce 12 for the denominator of the resulting 
fraction. 

Both terms of % multiplied by 4= 8... 258). 


Blackboard form. 
a ee! Si, 


3 <457-19 
153. Other Forms. 


Thought form for b. 

For the first equation we take 1=13. Equation (2) 
is found by dividing both members Be equation (1) by 
3. Hquation (8) is obtained by multiplying both mem- 
bers of equation (2) by 2: We thus find that = %;. 


Pe dias Ailes Se 


88 FRACTIONS. ye 


Example 2. Reduce § to a decimal ‘action. 

Blackboard form. 
7X125 875 Thought form. 
8125-1000 870. In the light of Prin. V, Z is 
tt veduced 10) a decimal epee 
plying both its terms by such a number as will pr>- 
duce a decimal denominator for the resulting fraction. 
Both terms of § multiplied by 125= 7,5. .. $=.875. 

Remarts. 1. A decimal fraction may be transferred from the 
fractional to the decimal notation by omitting the written denoin- 
inator and supplying the decimal point. 

2. A decimal fraction may be transferred from the decimal to 
the fractional notation by writing the proper denominator under 
the written decimal and removing the decimal point from the 
written numerator. 


3. The changeof noiation thus effected is not properly a reduc- 
tion, for the size and number of fractional units in the fraction 
remain unchanged. 

4. If a decimal fraction is to be reduced to smaller decinial 
units, the application of the above form is best shown by express- 
ing the given decimal in the fractional notation. [See ex. 4.] 


ea 3. Reduce .2 to thousandths. 
bse LOO 
for 000 100, 


and .2=2 times 100—=.200. 


Remark. The form of solution for example 3, is substantially 
that given for a under Art. 153. 
Example 4. Reduce .3 to hundredths. 


Thought form. 

The unit of the number to be reduced is .1. The 
unit of the number to be obtained is .01, a smaller nit 
than .1; hence reduction descending is required. 

For convenience in applying the principle, .3 may 


Mey 


Pe v 


« 


REDUCTION, 89 


be thought in the fractional form, 3;,. Multiplying 
both terms of 3 by 10, we obtain ;3,, as the required 
number, 
154. Case Ill. Fractions to a common and the 
last common denominator. 

memark., This case is one of reduction descending in which 
the reduction is effected as in Case II. 

Definitions. a. Fractions have a common denom- 

inator if composed of like fractional units. 


5. Fractions have their least common denomina- 
tor if composed of the greatest possible like fractional 


units, 

Remarks. 1. Before reducing fractions to equivalent fractions 
having the 1. c. d. it is necessary that the terms of each fraction 
be relatively prime. 

2. In Case II, the denominator of the fraction resulting from 
the reduction is a multiple of the denominator of the given 
fraction. From the nature of the method employed in effecting 
the reduction, this must always be the case; hence a common 
denominator of fractions must be a common multiple of the 
denominators of the given fractions. 

Principle. The least common denominator of given 
fractions is the 1. c. m. of their denominators. 

Example. Reduce 2, #and 2toequivalent fractions 
having their least common denominator, 

Thought form. 

(1. Prin. The least common denominator of given 
fractions is the least common multiple of their denom- 
inators. 

(2:) The 1. c. m. of 8, 4 and 6 is 12, hence each of 
these fractions must be expressed in 12ths. 

3.) $=) #=*%; t=1- 

(4.) .. 2, and 2 reduced to equivalent fractions 
haying their 1. c. d. equal, respectively, 3%, 7% and 4$. 


7 


“Rw Or) YAY a ae Lee f 
0? OMT tee fae 
’ Ll 4 é ae. 
au a NL 


90 FRACTIONS, 


REDUCTION ASCENDING. 


I55. Case I. A fraction to an integer or a mixed 
number. 


Example. 1. Reduce 42 to an integer. 
| First thought form. 

The unit .of the number to be reduced is 4. The 
unit of the number to be obtained is the unit 1, a 
greater unit than 4; hence reduction ascending is re- 
quired. By dividing both terms of +42 by 3 we obtain #, 


which equals 4, the required eee 


Second thought form. 

Since a fraction is reduced to larger units by divid- 
ing both its terms by the same number greater than 1, 
12 may be reduced to integral ones by dividing both 
its terms by 3... The result—4—4. Therefore, 12—4. 


Blackboard form. 
12484 
3+3 1 


156. Other Forms. 
Example 1. 41=what integer or mixed number? 


g—-1; 
Pe 
b | Li. Las many 1’sas 5 is contained times 


in 17, or 32. 
Pile Wey PES 
We wae 


Example 2. Reduce ? to the decimal scale. 


$—3-—4, 
Form.) bs ey. al 5 
$—.70. 


REDUCTION. , one 


Remarks. 1. The denominator of a decimal fraction is a power 
-of 10, the prime factors of which are 2 and 5, It follows, there- 
fore, that if a fraction (in its lowest terms) have in its denomin- 
ator a factor other than those of 10, such fraction cannot be 
reduced to a decimal. A common fraction is reducible to a 
decimal if the denominator of the given fraction is divisible by 
2 or 5, or by 2 and 5 and by no other prime number. If a fraction, 
in its lowest terms, have in its denominator a prime factor other 
than those of a decimal denominator, a repetend, or circulate 
will result upon attempting to reduce the fraction to the decimal 
scale. 

2. The reduction of a common fraction to a decimal fraction 
may be effected by either reduction. The guiding thought is— 
to multiply or divide both its terms by such a number as will 
give a decimal denominator to the resulting fraction. 


Reduce each of the following to a decimal scale or to 
a complex decimal. 


A. 
Bai ait B58) Ee) $; naka es 6? 
13-15. 3) Diente ut) he 4; 2. ATA Weary Pyare ss Bbaoumees te) 
16) 183903 203 203 9396969777) 77712) 7T7)7818- 
157. Case Il. A fraction to larger fractional units. 
Example. Reduce § to dds. 
First maa form. 


The unit of the number to be reduced is }. The 
unit of the number to be obtained is 4, a larger unit 
than 4; hence reduction PouOINS isrequired. — Divid- 
ing both terms of £ by 8 gives 2 as the required num- 
ber. ».*. $=2 

Second thought nal 

In the light of Prin. VI, § may be reduced to 3ds by 
dividing both its terms by sucht a number as will give 
8 for the denominator of the resulting fraction. Both 
terms of $ divided by 3,3. .. §=:4. 

Remark. A fraction is reduced to its lowest terms by dividing 
pon its terms by their g.c.d. Why? 

Blackboard form. 
ens ey. 
95 


OTe 
2) 9S ; 


ee Diree (Seis <> iy hag Oe Th scans 


92 FRACTIONS. 


Addition and Subtraction of Fractions. 


158. 


Principle—Only like units can be added. 

Principle—Only like units can be used in subtrac- 
tion. 

Remark. In adding or subtracting fractions it is to be ob- 
served that the numerators are the numbers with which we deal 
while the denominators only give name to those numbers. 

Example. Add 2,? and 2. 

Solution. Since only like units can be added, these 
fractions must be reduced to a common denominator 
before adding. | 


Blackboard form. 
208 48 Oye eee 
8°) 12,747 12 6 12. 


8 4/9 Grip a oF 
iat iota 12 


4h gmat, 


Remarks. 1. A form for subtracting one fraction from another 
is similar to that for addition. 


2. If mixed numbers are to be added or subtracted, the inte- 
gers and the fractions may be operated upon separately and the 
results combined, or, the mixed numbers may be reduced to 
fractions and the result found by the form given for examples 
in addition, and by a similar form for examples in subtraction, 


[ Exercises. | 


1. Add 4, 2 and 4, 1, From #ftake 2. 
2. “ and 4. 2 zh aug 3. 
3.“ 8,44 and 2. 3. if eae 4. 
4, “ 2 3 and $. 4, 6 i orl! ea 
One Bey alg, aio andG. 9 O. . S ee 


Sh a eo 


MULTIPLICATION. 93 


Multiplication of Fractions. 


159. Casel. To multiply by an integer. 


Example 1. Multiply »; by 3. 
Thought form. 
Since a fraction is multiplied by multiplying its 
numerator, 7% <3d=}}- 


Example 2. Multiply 33 by 2. 


Thought form. 


Since a fraction is multiplied by dividing its denom- 
inator, 73; <2==2. 


Remark. If the product should not be in its lowest terms or 
if it be reducible to an integer or mixed number it should be 
reduced. 


[ Exercises. ] 


160. Case ll. To multiply by a fraction. 
Example 1. Multiply 6 by 2. 
First form. 


6xX1=6. 

64-4 of 6=—2. a 
6x 2=2 times 2—4, 5 
“. 6X2—4. 


Second form. 
The multiplier = 4 of 2. 
Gea 12 
6X+4 of 2=-4 of 12—4. 


94 - FRACTIONS. | 


Third form. 
Principle. The product sustains the same relation 
to the multiplicand that the multiplier does to 1. — 


In this example the multiplier is 2 of 1; hence the 
product is 2 of 6. 


4 of 6=2. 
2 of 6=2 times 2=4. 
“. OxXg4, 


Fourth form. 
[Give principle and statement as in third form. ] 


2 of 6=+4 of 2 times 6, or 12. 


Example 2. Multiply. ? by #. 


Remark. The forms of solution for this example do not dif- 
fer from those given for example 1. Those forms are, however, 
applied to the solution of example 2. 


First form. 
813 
aad Roe 3 
a X77 of gas 
$x $5. times 3,18. 
Pee pete 
Second form. 
The multiplier is + of 5. 
$x5=15 
$ x4 ner 152-138 
8X4 of 5=+ of 18=23. 
3x OY 
4 ts f 2 8° 


MULTIPLICATION. 95 


Third form. 
Principle. The product sustains the same relation 
to the multiplicand that the multiplier does to 1. 
In this example the multiplier is # of 1; hence the 
product is # of #. 


Alco 
x 
aon 
|| 
bol 
cojen 


Fourth form. 
[Give principle and statement as in third form. ] 
| 5 of 8=1 of 5 times 3, or 4,8, 
bof =f, 
EXT Te 
Fifth form. 
(1.) $X4=8. 
(2.) #X7=6. 
(3.) $X#X28=15. 
(4) =X7=t- 
iB UC. 


Remark on fifth form. The multiplicand multiplied by its 
denominator equals its numerator. 

The multiplier multiplied by its denominator equals its numer- 
ator. Multiplying together the corresponding members of these 
equations gives equation (3); and dividing both members of 
equation (3) by 28 gives equation (4). 

The first member of equation (4) consists of the two factors 
whose product is required, while the second member is the 
product sought. 


[S!. General Remarks on Multiplication of Fractions. 


1. The product of one fraction by another may be obtained 
by multiplying together the numerators of the factors for the 
numerator of the product, and the denominators of the factors 
for the denominator of the product. [See rule in any good text 
hook on arithmetic. ] fe 


96 FRACTIONS. 


2. Mixed numbers may be reduced to fractions before multi- 
plying; or the multiplicand may be multiplied by the integral 
and the fractional parts of the multiplier (used separately) and 
the partial products combined. 

3. If one or both factors be decimal fractions expressed in 
the decimal notation, any of the givenforms may beused. The 
second form is, perhaps, the best to use in the multiplication of 
decimal fractions. Its use is illustrated in the following 

Example. Multiply 2.5 by .025. 

Form. The multiplier equals .001 of va 
2.5 < 20 =o 2, 
2.5%X.001 of 25001: of 62.5-—.0695, 
Sod KE U LOS Ose 


[ Exercises. ] 


Division of Fractions. 
162. Casel. To divide by an integer. 
Example 1. Divide $ by 3. 
Thought form. 

Sincea fraction is divided by dividing its numerator, 
$+-3=2. 
Example 2. Divide $ by 5. 

Thought form. 

Since a fraction is divided by seheeg stir its denom- 
inator, $+5=,%. 

[ Exercises. | 


163. Case Il. To divide by a fraction. 


Example 1. Divide 5 by #. DMR SHS Sion 
First form. . 
an t==5, 


Whe times 535. 
ea -3=—=4 of 35=112. 
Hee Aare Is 


DIVISION, 97 


Second form. 
The divisor equals 4 of 3. 


5+-3=-3 
5+4+ a Nehaal times 3=35—114. ) 
* 0+8=112. 
Third form. 
1=-4=7. 
1--3=4 of 7=}. 
5+#=—5 times {=33—=112. 
*. 6+ 3112. 


Example 2. Divide $ by 2. 


Remark. For the first three forms see those given for the 
preceding example. 


, Fourth form. 


Thought form. 


The dividend equals itself multiplied by the divisor 
multiplied by the reciprocal of the divisor. 

Dividing both members of equation (1) by 3 gives 
equation (2). the first member of which is the indi- 
cated division to be performed while the second mem- 
ber. in its simplest form, is the required quotient. 


93 FRACTIONS. 


164. General Remarks on Division of Fractions. 


1. One fraction may be divided. by another by dividing 
the numerator of the dividend by the numerator of the divisor 
and the denominator of the dividend by the denominator of the 
divisor. bast 

If the terms of the dividend be not respectively divisible by 
the corresponding terms of the divisor, they may be made so 
by multiplying both terms of the dividend by the product of 
the terms of the divisor. 

Division of fractions is thus shown to be the reverse of multi- 
plication of fractions. 

2. If the divisor isa mixed number, it is, perhaps, better to 
reduce it to a fraction before dividing. 

3. If the divisor is written wholly in the decimal notation, 
the above forms are as applicable as though it were written in 
the fractional notation. The following form is preferred. 


Example. Divide .0625 by 2.5. 
Form. The divisor is .1 of 25. 
.0625--25—=.0025 
.0625-—.1 of 25=10 times .0025=..025. 
»*. .0625-—-2.5==.025. 


[ Exercises in division. | 


165. Exercises Involving Fractions. 
1, 2 bu.==what part of 3 bu? 
Solution. *.. 1 bu.=$ of 3 bu, 
2 bu.=2 times 4 of 3 bu., or 
2 of 3 bu. 
sa DU.s=eot 3 bu. 


2. $4—what part of $10? 

3. 15 apples=what part of 50 apples? 

4, 4 acre—what part of 4 acres? 

5. 2 of a gallon=what part of 5 gallons? 
6. 2 of a bu.=what part of 10 bu.? 


7 
« Ohh Ae 
Neel a ae 


EXERCISES. | 99 


7. 2 of an orange=what part of ? of an orange? 
Solution. 
4 of an orange=4 of ? of an orange, 
+ of an orange—4 times 4 of 2 of an orange, or 
4 times # of an orange; and 4 of an orange 
=+4 of $ times 3 of an orange, or § of 3 of an 
orange; and # of an orange—2 times ¢ of 3 
of an orange, or $ of # of an orange. 
.. # of an orange—$ of 3 of an orange. 
8. what part of #? 
. 2==what part of £? 
10. 3=what part of 3? 
11. =what part of 3? 
12. what part of 3? 
13. How much water will fill 4 tubs if each tub 
holds 54 gallons? 
14. What cost 9 apples at 14¢ each? 
15. If 8 lb. of sugar sell for $1, what is the price per 
pound? 
16. At $6 per cord, required the cost of 2 of a cord 
of wood. 
17. Required the cost of # lb. of peaches at 35¢ per 
pound. 
18. If coal is $3 per ton, required the cost of 43 tons, 
19. If a train run 15 miles per hr., how far will it 
run in 34 hours? 
20. At $2 per bushel, required the cost of % of a 
bushel of corn. , 
21. What cost # of a gallon of syrup, at $14 per 
gallon? 
22. If a man cut 3 of a cord of wood in a day, how 
much can he cut in 4 of a day? 


100 FRACTIONS. 


23. A boy divided +, of a bushel of apples among 4 
playmates; what part of a bushel did each receive? | 

24. A man had 2 of a barrel of pork and sold of it; 
what part of a barrel remained? ¢ 

25. A man lost 4 of his money and found $ as much 
as he had after his loss; what part of his original sum 
had he then? 

26. If 1 yard of ribbon cost $,3,, how many yards can 
be bought for $3? : ; 

27. How many yards of cloth will $10 buy at $23 per 
yard? 

28. If 2 men doa piece of work in 43 days, how 
long will it take 8 men to do it? 

29. What cost 31 boxes of oranges, if 24 boxes cost 
$9? | 

30. What cost 30 bushels of corn, if 34 bushels cost 
$1.20? | 

31. If a bushel of wheat cost $4, what cost 8 of a 
bushel? 

32. Ifa bushel of wheat cost $4, how many bushels 
can be bought for $5.20? 

33. A girl divided 10 apples among her companions, 
giving to each 2 of an apple; how many companions 
had she? 

34. 4=2 of what number ? 

30. # of 14=—¢ of what number? 

36. 8 of 36=% of what number? 

37. 2 of $40=2 of the cost of a horse; required its 
cost. | 

38. After spending + of his money, John had $42 
remaining; how much had he at first ? 

39. If # of an acre of land be worth $15, what are 12 
acres worth? 


EXERCISES. IOf 


40. A boy sold lemons at the rate of 6 for 8 cents ; 
how much did he receive for 3 lemons? For 8 lemons ? 
For 12 lemons? | 

41. If 2 of a yard of silk cost $31; what cost 4} 
yards? 

42. If 2 of a yard of cloth cost $4, what cost 8 of a 
yard ? 

43, A man gained $15 by selling a watch for 1? times 
its cost; required its cost. 

44. Mary, after losing 2? of her flowers, had but 3 
remaining ; how many had she at first ? 

45. If to $ the cost of John’s coat $10 be added, the 
sum is $21; required its cost. 

46. If to of William’s age 8 years be added, the 
sum is 1} times his age; how old is he? 


47. Two men hire a wagon for $9; A uses it 7 days, 
and B uses it 2 days; what should each pay ? 

48. John and James bought 22 apples for 11 cents ; 
John paid 7 cents while James paid 4 cents; how many 
apples should each receive? 

49. Anna has 5 pinks more than Ruth, and together 
they have 19; how many has each ? 

50. A boy said that 4 is 3 less than 4 his number of 
marbles ; how many has he? 

51. 10 years are 6 years more than ? of John’s age ; 
how old is he? 

52. If A and B do 33, of a piece of work in a day, 
how long will it take them to do the entire work? 

538. George can plow a field in 8 days, and Henry 
can plow it in 12 days; how long would it take them 
to do the work, working together ? 

54, If ¢ of a barrel of flour cost $43, what cost $ via 
barrel ? 


102 FRACTIONS, 


55. If A can do 2 of a piece of work in a day, how 
much can he do in 2 days? 

56. What cost 6 bushels of glo seed, if 2 bushels 
cost $123? 

57. If 38 ofa OnE of coffee cost 97, what cost $ of 
a pound ? 

58. A has $13 which is 2 of twice as much as B has; 
how much has B? 

59. A horse is sold for $60 which is $ of 3 of its value ; 
required its value. 

60. Henry and George bought 30 nuts; Henry paid 


18 cents and George paid 12 cents; how should the nuts 


be divided ? 


61. A man failing in business can pay 40 cents on the 
dollar ; what part of his debts can he pay ? 

62. 4? of what number ? 

63.. A has $3 more’than B; and together they 
have $74; how many dollars has Pek z 

64. 42 of twice what number ? 

65.. If 4 of a box of berries cost $f, what cost 2 of a 
box? 

66. What is the number if its + increased by 10 
equals 21? 
_ 67. Required the number if its + added to its 4 
equals 3. 

68. 2 of a number ++-5=26; what is the number? 

69. A farmer sold 4 of his grain, and had 120 bushels 
remaining; how much had he at first? 

70. Required the cost of 15 horses at the rate of 3} 
horses for $169. 

71. How much will 33 acres of land cost at $64 for 
1} acres? 


EXERCISES. 103 


72. The difference between 2 of a numberand $ of it 
is 6; what is the number ? 

73. If 25 times a number exceed 2 times the number 
by 32, what is the number? 

74. If a man walk $ of a mile in 102 minutes, how 
long will it take him to walk 5 miles? 

7). Required the cost of 4 dozen eggs at 25 cents for 
10 eggs 

76. Required the cost of 45 apples at 10 cents per 
dozen. 

77. $6=2 of 4 of a sum of money; required the sum. 

78. How many yards of goods will make 3 dresses 
if 15 yards make 3 of a dress? 

79. 2 of the length of a pole is in the water and 154 
feet-are out; what is the length of the pole? 

80. If 19 boxes of berries are worth 57 cents, required 
the value of 2 of a box. 

81. If 2 of a yd. of cloth cost $13, what cost 22 yd? 

82. 2 of John’s money equals # of Harry’s, and 
together they have $55 ; how much has each? 

83. If on 1 orange I lose 3, of a cent, how many 
oranges must I sell to lose 6 cents? 

84. William has twice as many cents as Herbert, and 
together they have 24 cents, how many has each ? 

85. Two boys have 49 marbles; one has 7 more than 
the other; how many has each ? 

86. Henry received for his horse ¢ of its cost; what 
part of the cost was the gain ? 

87. A coat which cost $12 was sold for $16; the gain 
was how many hundredths of the cost ? 

88. Goods bought at $12 were sold for $10; how 
many hundredths of the cost was the loss? 


104 FRACTIONS, 


89. A boy bought some apples for 72 cents and sold 
them for 84 cents; the gain was what part of the cost ? 
How many hundredths of the cost ? 


90 A horse was bought for $60 and sold for $48; what 
part of the cost was the loss? How many hundredths 
of the cost was the loss ? 


91. # of $6=how many hundredths of $20? 


92. For what must goods costing $50 be sold to gain 
ten hundredths of the cost? 


93. A paid $80 for a horse and sold it so as to lose 
rey of its cost; for what did he sell it ? 

94. If a merchant sold goods at $2 per yard and 
thereby gained 2,5, of the cost, required the cost. 

95. A cow was bought for $25 and sold for $30; what 
part of the cost was the gain ? 

96. Henry can make 2 of a pair of boots in a day, 
and James can make # of a pair in a day; how long 
will it take both to make 2 pairs of boots? 

97. In how many days can 3 men cut 15 cords of 
wood, if 1 man in 1 day cut 2 of a cord? 

98. A boy bought a certain number of apples at 2 
cents each, and the same number at 4 cents each, and 
then sold’ out at the rate of 8 for 5 cents; did he gain 
or lose and how much? 

99. If ? of an orange cost as much as } of a pineap- 
ple, required the price of two oranges in pineapples. 

100. A man gained $10 by selling his horse for 14 
times its cost; what was the cost? 

101. A can do a piece of work in 5 days and B can 
do it in 8 days; in whaf time can they together do it? 


102. A lumber dealer bought siding at $18.75 per M | 


and sold it at $2.875 per C; how much did he gain 
per M? 


my 
é aah 


, a 


ALIQUOTS. * 105 


Aliquot Parts: 


166. Table of Aliquots. 
| =} of 10, 183= 3, of 100. 


3i=1 of. 10. 20 =+ of 100. - 
64—,, of 100. 20 ==+, of 100. 
+ §1=,, of 100. d0t= 4 of 100. 
2g = OF 10), 624= 2 of 100. 
162= 3 of 100. 662— 2 of 100. 


125 = 4 of 1000, 


167. Forms of Solution. 


Example 1. At 18? cents per lb. required the cost 
of 32 lb. of butter. 

Solution. At $1 per lb. 32 lb. of butter cost *32, but 
at 183 cents, or $33; per lb, 32 lb. cost 3% of Pe: or $6. 
etc. 

Example 2. At 124 cents Be lb. how many lb. -of 
rice will $24 buy ? 

Solution. At 124 cents, or gL per lb., $1 will buy 8 
lb., and $24 will buy 24 times 8 lb., or 192 Whe rete: 

Example 3. . At $874 per acre, how many acres ‘of 
land can be bought for $4900 ? 

Solution. At $100 per acre, $4900 will buy 49 acres, 
but at $873, or g of $100, $4900 will buy 8 times 4 of 49 
acres, or 56 acres, 


Exercises. 
1. At 64¢ each what cost 16 oranges ? 
2 At $6.25 per barrel what cost 7 barrels of flour? 
d. At 24¢ apiece what cost 20 pencils ? 
4. At $2.50 a box what cost 60 boxes of potatoes? 
5. At 84¢ per yd. what cost 56 yd. of muslin? 
§ 


FRACTIONS. 


. At $8.334 each what cost 30 calves ? 

. At 124¢ a lb. what cost 64 lb. of sugar ? | 

. At $12.50 per acre what cost 17 acres of corn? 
. At 162¢ per lb. what cost 40 lb. of butter? 

. At 162¢ each what cost 4 slates ? 

. At 183¢ per lb. what cost 32 lb. of steak ? 

. At $18.75 each what cost 80 ponies ? 

. If 1 doz. eggs cost 20¢ what cost 15 doz.? 

. At 20¢ each what cost 7 books? 

. If 1 cow cost $25, required the cost of 16 cows. 
. At 25¢each what cost 28 collars ? 

. At 34¢ apiece what cost 9 apples? 

. At $3.50 each what cost 15 hats? 

. At $334 per acre what cost 30 acres of land ? 

. At 50¢ each what cost 7 books ? 

. At 624¢ each what cost 17 pitchers? 

. At $662 per head what cost 12 horses? 

. At $1.25 per rod what cost 10 rods of fencing? 
. At $125 per head what cost 14 horses? 

. At $20 per acre what cost 15 acres of land ? 

. At 374¢ per yd. what cost 7 yd. of silk cord ? 
27. 
28. 


At $75 per head required the cost of 12 horses. 
At 64” per spool required the cost of 11 spools 


of thread. 


CHAPTER VIII. 
COMPOUND NUMBERS. 


168. Compound numbers are classified on the basis 
ef the kind of attribute measured, as— 

1. MEASURE OF DURATION. 

2. MEASURES OF EXTENSION. 

3. MEASURES OF FORCE. 


169. Diagram exhibiting in classified form the “ Meas- 
ures” usually treated under compound numbers. ' 


Remarks, -1. The decimal measures are embraced in the dia- 
gram, but are treated separately. 

2. The measure of value in most civilized countries is derived 
from the force of gravity, or weight. The primary units of value 
were weight units. 


1. Of Duration.—Time measure. 


Long measure. 

Surveyors’ long measure. 
Length + Mariners’ “ 

Decimal 

Circumference 


6c éé 
6 


Square measure. 
2. Of Extension / Surface + Surveyor’s square : 
Decimal " 
Measures 


Volume + Dry measure. 
Liquid measure. 


Decimal capacity “ 


Troy weight. 
Apothecaries’ weight. 
Avoirdupois ec 


Decimal 


3. Of Force. 
United States money. 
English 

Ete. 


Cubic measure 
Decimal cubic measure. 
Value 


ae ee 


108 COMPOUND NUMBERS. 


170. Order of Study. — Ra AIH 


temark. The following order shou!d be followedin the study 
of each measure. 
~- “(1.) The primary, or standard unit How deter- 
mined. ? ae 
(2.) Other units and their relative value. 
(3.) Scale and table.. 
(4) Reduction. 
a. Descending. 
1. Integers to integers of low er denoi mu nation, 


2. ractions to e rf 
>, <3 ‘* fractions of ‘ 


66 


b. Ascending. 


1. Integers to integers of higher denomination 


ape a (a3 fractions of (a4 fa . 
3. Fractionsto ‘* 6b be ‘c 


(5.) Synthesis. 
a. Addition. 
b. Multiplication. 
(6.) Analysis. 
a. Subtraction. 
b. Division. 
171. Tables. 


For the tables of compound numbers and many 
interesting and useful facts the pupil is referred to text 
books on arithmetic and to encyclopedias. 

Applications of Compcund Numbers. 


Remark... The “ order of study” will be applied. in part, to 
‘time measure.” Each of the other measures should betreated 
in a similiar manner. 


172. Time Measure. 


Time. That which renders succession 1 posite is 
called tzme. 


or 4 
eel Ft > 
i oe ee i 


RELATIONS. 109 


The Primary Unit.. The average solar day is taken 
as the primary unit of time. 

(1.) A Solar day is the interval of time between 
two successive: transits of the vertical ray of the sun 
across a given meridian. This interval varies at dif- 
ferent times of the year. 

(2.) The sidereal day is the interval between two 
successive transits of a fixed star across a given merid- 
ian. This interval is the same at one time that it is at 
another. 

(3.) The solar and the sidereal days would be of 
equal length if the earth did not revolve around the 
sun. While the earth is rotating upon its axis it is 
also moving forward in its orbit; so that when it has 
made a complete rotation, it must make part of another 
before the sun’s rays are vertical a second time upon 
any given meridian. The solar day is thus a little 
longer than the sidereal day. [About four minutes. ] 

Other Denominations. The other time units are either 
multiples or divisors of the day. 

The multiples of the day are the week, the month, 
the year and the century. 

The divisors of the day are the hour, the minute and 
the second. 

; Relations. 

Multiples. 

1. The week equals 7 days. 

2. The month equals 42 weeks. 

Remark, The average calendar month is a little more than 44 
weeks, or 30 days, while the lunar month is a small fraction 
more than four weeks. 7 

do. The year equals 12 calendar months. 

4. The century eyuals 100 yea’s. 


LID? COMPOUND NUMBERS, 


Divisors. 
1. The hour equals 3 of the day. 
2, The minute equals 1, of the hour. 
3 The second equals 4 of the minute. 


Scale. The units used in time measure may be writ- 
ten in a scale as follows: 


100 19....'43. 7 04 a 
cen.) yr. -mo.-. wk. 'da’ hr aie 


1 1 1 1 1 a 1 1 


Remarks, 1. The values expressed by the respective units of 
this scale increase from right to left in the written scale. In 
this respect the time scale is like the decimal scale. 


2. Since the rate of increase varies, the time scale is called a 


varying scale. In this respect it is unlike the decimal scale, 
whose rate of increase is uniformly 10. 


_3. The time scale consists of but eight units. In this respect 
it is unlike the decimal scale, whose orders of units may be 
repeated in periods indefinitely 


4. In the time seale the units extend both above and below 
the primary unit. In this respect it is like the decimal scale. 


Table. For convenience the relations of the time 
units may be tabulated thus: 


60" see; Ss ins 
GOs v be: 
DAL ya layer bela 
Tp Wed eae al aR 
42 wk. = 1 mo. 
12*mot ie yet 
LOO! yr | = Liecen. 


a 


REDUCTION. 


Reduction. 
173. Reduction Descending. 
Example 1. Reduce 2 wk. 3 da. 12 hr. to min. 
First form. 
aids Wa oe ROA, 
Be ieee tMmes: (Cad da, 
14 da.+3 da. == 17 da. 
meas a= 2e hy,: 
LP == 37 times 24:hr. == 408 hr. 
408 hr.+-12 hr. = 420 hr. 
Beek wee. GOomin., 
420 “ = 420 times 60 min.=25200 min. 
“.2wk. 3da. 12 hr. = 25200 min. 


Second form. 
Sok Wk == 7 times’ 1 ‘da., 


My Meee A Rh celp re kc LCL 
14 da.+3 da. = 17 da. 
een as 24 times Lihir,, 
Ti theh cass BE A F408" hr, 
408 hr.4-12 hr. = 420 hr. 
*- 1 “ — 60 times 1 min., 
BA ace O03) 420i °° —=25900 min: 


te 2 Wikio Ga. 22 nr. 25200'min. 


Example 2. Reduce 3 wk. to smaller denominate 
units. 


First form. 
Sa wk i=1. da.; 
==§ of 7 da.=38 da. 
a Lida.ca24 whr., 
==$ of 24 hr.=214 hr. 
oaks Ot) Min), 
==1 of 60 min.=20 min. 
. 2 wk.=8 da. 21 hr. 20 min. 


112 COMPOUND NUMBERS. 
Second form. 
1 wk.=7 times 1 da., - Nate Oh 
7 ‘* =7 times $ da. 38 da. Ki 


oe aaa oh 1 hry 

Shs Dd tA Ra ee as 
Dre Ou Cyn 1 Min; 20pe 

“ =360 * 2 8 ==20, min. 
wk. =3 da. 21 hr. 20 min. 


° e 
, e 
Col ciao 


“3 
Erample 3. Redes +2, of a wk. to the fraction of 
an hour. 


First form. 
 bwkesjedas 
Too | = 77 Of 7 da.—.03 da. 
Lida eam. 
OBES ree of 24 hr.=33 9 
ribo wk =3$ hr. — 


Second form. 
°° 1 wk.=7 times 1 da., 
rh peed eae a se geenn) OCs 
‘1 da-—24 hrs 
03.2) sa2455 08 18 We 
weg Wk.=H8 hr. 


Remarks. 1. Two forms of solution have been given for a: 


problem in each case under reduction descending The first 
form in each case is that usually given for such problems and 
needs no comment other than the observation that the multi- 
plier is not taken from the table but is the pS (taken 
abstractly) to be reduced. 


2. 'The second form rests upon the following:— 
PRINCIPLE.— The numerical relation that exists between 
given units exists between like multiples and also between like 
parts of those units. 


| ... REDUCTION. - at 


| 3. The first step consists in the statement of the relation exist. 
ing between a unit of the denomination to be reduced and a _ 
unit of the denomination to which the reduction is to be made. 
The second step is made‘in the light of the above principle; e. 
g., Since 1 hr. = 60 times 1 min., 420 hr. (a multiple of 1 hr.) 
equal 60 times 420 min: (a like multiple ofl min) Itis observed 
that the multiplier is, in every instance, taken from the table. 
This form of solution is uniform and general in its application 
to the solution of all problems in the several cases of reduc- 
tion descending in all the measures. 

4. A careful study of the second form given for the solution 
of problems in reduction ascending, will show the form to be 
uniform and of general application in all the measures. 


5. Any reduction, either descending or ascending, may be 
made by a direct use of the equation. ‘The first statementunder 
the second form in each of the given examples is taken as the 
first equation. The second equation is obtained from the first 
by multiplying both its members by such a number as will give 
the number to be reduced for the first member of the second 
equation. In transforming the first equation it is to be observed 
that its second member consists of two factors, and that the 
member is to be multiplied by multiplying its second factor. 


174. Reduction Ascending. — 
Example 1. Reduce 25200 min. to integers of higher 
denomination. 


First form. 
*. 60 min =1 hr, 
25200 ‘“* as many hr. as 25200 min. 


are times 00 min. which=420. 
724 hr.—=1 da., 
420 hr.=as many da. as 420 hr. are 
times 24 hr. which=17, with a remain- 
der of-12 hr. 
Heda 1:wk., 
17 da.=as many wk. as 17 da. are 
times 7 da. which=2, with a remain- 
» =der of 3.da.... 
». 25200 min.=2 wk. 3 da. 12.hr. 


114, COMPOUNL) NUMBERS, 


Second form. 
“1 min. =a of Thr. 


95200 A =e of 25200 hr,=420 hr. 
~ 1 hrs) of 1 da., 
420 “ = of 420 da.=17 da. 12 hr. 


ee wie “Gf 1 wk., 
s ty tee OE Lh ve =? wk. 8 da. 
*, 25200 min. =) wk. 3 da. 12 hr. 
Example 2. Reduce 8 da. 21 hr. 20 min. to the frac- 
tion of a wk. 
First form. 

OO) gy sak Eee, 
20 ‘“ =as many hr. as 20 min. are 
times 69 min. .which=4., 
21 hr,+4 hr.=214 hr, 

‘24 hr. ==1 da., 
214 hr. =as many da. as 214 hr. are 

times 24 hr. which=8, 

3 da +£ da.=88 da. 

A a= eWay s 
38da.=as many wk. as 3§ da. are 

times 7 da. which=32. 
*,3 da. 21 hr. 20 min.=3 wk, 
Second form. 
** 60 mint==t hr, 


20 ‘ ==such part of 1 hr. as 20 is part of 60 
which is 4. 
21 hr.+4 hr.=214 hr. 
O24 nrecd da., 
214 hr.=such part of 1 da. as 213 is part of 24 
which is $. 
3 da.t$ da.=3§ da. 
et Le Cheetos) LAW Ret 


38 da. such part of 1 wk. as 38 is part of 7 
which is 3 _..*. etc. 


REDUCTION, 115 


Third form, 

** 1 min.=-,; of 1 hr., 

20 “ =, of 20 hr.=4$ hr. 

21 hr.+4 hr.=214 hr.—§¢ hr, 
°° 1 hr.==,, of 1 da., 

$4 hr.= 5, of §4 da.=$ da. 

3 da.t+8 da,=38 da.=% da, 
*° 1 da.—+ of 1 wk., 

$5 da.=+ of 3 wk. wk. 
”. d da. 21 hr. 20 min.=8 wk. 


Remark. The several denominate numbers constituting the 
given compound number, inay be reduced to the lowest denom- 
ination (min.); and this number of minutes may then be 
reduced to the fraction of a week. Mixed numbers are thus 
avoided. 


Example 3. Reduce }% hr. to the fraction of a wk 
First form. 
[Use either the first or second form given under Ex- 
ample 2.] 
Second form, 
‘1 hr.=,, of I da., 
48 hr.=,; of 4% da.—.03 da. 
*’ 1 da.=+ of lwk., 
.03 da.==4 of .03 wk.=73,5 wk. 
. 48 hr.=-3y wk. 
175. General Remarks. 
1, Such examples as the following are often given: 
2 da.5 hr. 15 min. equal what part of 2 wk. 4 da. 
3 hr.? 


In such cases each compound number should be 
reduced to the lowest denomination given in either, 


1€6: COMPOUND. NUMBERS. 
Solution. ER eee 
2 da. 5 hr. 15 min,=3195 ‘min, © = #4 
2 wk. 4da. 3 hr.=26100 min. 
213 


‘1 min.=:53455 of 26100 min., 
8195 min.—=3195 times Io 100 SE 26100 min. — Pres 


of 26100 min. 
*. ete. 


2. The following two solutions are given for the purpose of 
exhibiting a method that shall obviate the use of a complex 
fraction in stating the relation of the less unit to the greater in 


each example. 


Example 1. Reduce 33 yd. to rd. 
The first step is to state the relation between 1 yd. 


and 1 rd., thus: 
I ydj=-4, of 1 rd., ete. 


. Since the ER ene as stated cannot be read as a frac- 
tion, it is well to express the relation Rhee Ti ae is 


the equivalent of 2. 


The solution should be as follows: 


ie 1 yd. = #7, of 1 rd., 
FOL Bose ae ae 33 rd. = 6rd. 
(ratB3 iit Ss Baa. 


Exvample 2. Reduce 132 ft. to rd. 


(Lf = grof Led. 

7 (4 9 6 

Lorn 132 ch Re of 32 rae = 8 rd. 
6) ie ASO eS Bird 


REDUCTION: 3-0 wl-7 


3. Addition, multiplication, subtraction and division of com- 
pound numbers may be effected in the same manner and in 
obedience to the same principles that govern the synthesis and 
the analysis of numbers expressed in the decimal scale. No 
new principle is introduced and but one new fact, viz.: a vary- 
ing scale is employed instead of a uniform scale. a 

4. The subject of time is placed first in the classification of the 
measures. because the primary units of.extension ae weight 
are derived from a time unit. . 

The primary unit of the common measures of extension is the 
yard, which is a definite portion of the length of a pendulum 
that beats seconds under certain conditions. 

The primary unit of weight is the pound which is the force of 
gravity that acts upon a certain volume of water under certain 
conditions. The primary unit of extension is derived directly 
from a time unit (the second), and the primary unit of weight 
is derived directly from a measure of extension (a yolume) and 
through that from the same time unit (the second.) 

The primary unit of noave is a certain weight of Son or 
gold. 


[Norr.—Circumference measure and the decimal (metric) 
measures are exceptions to the above remark. ] 


Exercises. 
. Reduce 5 wk. 3 da. 4 hr. to min. 
. Reduce 4 da. ?s hr 7% min. to sec. 
. Reduce 2 bu. 3 pk. 5 qt. to pt. 
. Reduce 2 bu. 2 pk. to qt. 
. Reduce 5 yd. 2 ft. 7 in. to in. 
Reduce 60 acres to sq. ft. 
. Reduce 3 hhd. 24 gal. 3 qt. to gills, 
. Reduce 4 lb. 15 pwt. to grains. 
. Reduce 2 oz. 2 pwt. to grains. 
. Reduce 73.23 to grains. 
. Reduce é lb. Troy, to integers of lower denomina- 
tions. 


SS CO260) STS Or Ha 


+ pu 
— 


118 COMPOUND NUMBERS. 


12. Reduce .3 da. to integers of lower denomina- 


tions. 

18. Reduce # yd. to integers of lower denomina- 
tions. 

14. Reduce .875 gal. to integers of lower denomina- 
tions. eh 

15. Reduce 75, da. to the denomination minutes. 

16. Reduce .007 gal. to the denomination pints. 

17. Reduce #, yd. to the denomination inches. 

18. Reduce 3 bu. to the denomination pints. 

19. Reduce 54960 min. to integers of higher denomi- 
nations. 

20. Reduce 186 pt. to higher denominations. 

21. Reduce 57648 sec. to higher denominations. 

22. Reduce 354 qt. to higher denominations. 

23. Reduce 211 inches to higher denominations. 

24. Reduce 23400 grains to Troy integers of higher 
denominations. 

25. Reduce # of a ft. to the fraction of a rd. 

26. Reduce § of a dr. to the fraction of a lb. 

27. Reduce 600 gr. to a compound number. 

28. Reduce 8211 oz. to acompound number of higher 
denomination. 


CHAPTER IX. 
TIME AND LONGITUDE. 


176. The longitude of a place on the earth is the dis- 
tance of its meridian, east or west from an assumed 
meridian.—[ Peabody. ] 

Remarks. 1. Longitude is estimated in units of circumference, 
yiz.: degrees, minutes and seconds. 


2. Unless otherwise designated, the meridian of Greenwich 
is taken as the prime from which longitude is reckone |. 


177. Because of the daily rotation of the earth on its 
axis, any place on the surface of the earth except the 
poles, moves— 


In 24 hours through 360° of space. 
66 1 hour 66 15° 3 
oy ei k, PON. . a 
oti x Tove i Lo 


Hence if 15 units of longitude lie between two places. 
the time registered at the places, respectively, differs 
by 1 time unit; hours corresponding to degrees, min- 
utes to minutes, and seconds to seconds. 

On the other hand, if the time of two places is found 
to differ by 1 time unit, the places are distant from 
each other (east and west) 15 corresponding longitude 
units. 


120 TIME AND LONGITUDE. 


All places on the earth’s surface have the same abso- 
lute time, but not the same relative time. 

When the vertical ray of the sun reaches the merid- 
ian of a place it is noon at all places on that meridian ; 
while it is afternoon, or later in the day, at all places 
east, and before noon, or earlier in the day, atall places 
west of the given meridian. Thus a place east of a 
given meridian has later relative time and a place 
west has earlier relative time than that on the given 
meridian. 

178. Standard Time. 

For the purpose of estimating time for the railway 
service, the country of the United States is divided 
into belts or strips marked by meridians of longitude 
15° apart. The 75th meridian from Greenwich marks 
the middle of the eastern belt The 90th meridian 
marks the middle of the central belt. The 105th meri- 
dian marks the middle of the mountain belt. The 
120th meridian marks the middle of the Pacific belt. 

The local time on each of these meridians is taken 
as the “Standard” time for railway purposes at all 
places within the given belt; andsince these meridians 
are 15° apart, the time in each belt is registered one 
hour earlier than in the belt next at the east; e. g., 
when it is noon on the 75th meridian, itis noon at all 
places within the eastern belt, 11 a. m. on the 90th 
meridian and at all places within the central belt, 10 
a.m. on the 105th meridian and at all places hats 
the mountain belt and 9 a.m. on the 120th. meridian 
and at all places within the Pacific belt. } 

Remark. In each belt near its prime meridian workshops, 
manufactories, public schools and many other branches of bus- 
iness are now generally carried on by “ Standard” instead of 
local time. 


«= f) os ek as = * 


SOLUTIONS. 7 121 


179. Forms of Solution. 


Example 1. The time between two places is 6 hr. 
12 min. 10 sec.; required their difference in longitude. 
Form. 
-- 1 hr. 1 min. 1 sec. correspond to 15 times 1° 1’ 1", 
6 hr. 12 min. 10sec. correspond to 15 times 6° 12’ 10” 
=-J5" 2.30". 
-. their difference in longitude is 93° 2’ 30”. 
Example 2. The difference in longitude between 
New York and Greenwich is 74° 3”; required their dif- 
ference in time, 


Form. 
-1° 1’ 1” correspond 4 zs of 1 hr. 1 min. 1 sec., 
74° 3”. : ce “74 3 ‘sec.=4 hr. 56 


Byih..25eC.-©.*. etc. 
Example 3. When itis noon at New York, what 
is Greenwich time? 
Remark. Their time difference is given above. 
Form. 
Since their time difference is 4 hr. 56 min. .2 sec., 
and since Greenwich is east of New York, its time-is 


later than that of New York by 4 hr. 56 min. .2 sec., 
or 56 min. .2 sec past 4p. m. 


Example 4. Wheu it is noon at Greenwich what is 
New York time? 


Remark. Their time difference is given above. 


a 


Form. 

Since their time difference is 4 hr. 56 min. .2 sec., and 
since New York is west, its time is earlier by 4 hr. 56 
min. .2 sec., or 3 min. 59.8 sec. past 7 a. m. 

oer ‘ | 
4 


5 
é 


te ei ee! tr oi aid tet oe 
.s Peak \Ponee Sa 


Pe TIME AND LONGITUDE. 


Exercises. 


1 Two places differ in time 6 hr. 7 min. 10 sec; 
required the difference in longitude. 


2 Two places are distant from each other 1 quad- 
rant; what is their time difference? If it is10 a. m.; 
at the place the farther west what time is it at the 
other? When it is 1 o’clock p.m. at the place the 
farther east what time is it at the other? 


3. A man travels westward 5° 17’ 11”; is his watch 

too fast or too slow and how much? 
+4. A man travels until his watch is 19 min. 55 sec. 
too fast; has he traveled east or west, and how far? 

5. A man travels until his watch is too slow by 11 
hr. 5 min 16 sec.; has he traveled east or west, and how 
far ? 

6. The longitude of St. Louis is 90° 10’ W., while that 
of Cincinnati is 84° 26’ W.; required their difference in 
time. Whenit is noon at either place what isthe time 
at the other? 3 


7. The longitude of Bangor, Me., is 68° 45’ W., and 
that of San Francisco is 122° 25’ W.; required their time 
difference. When it is 7 a m. at either place what 
time is it at the other? 

8. A vessel sailed due north at the rate of 14 knots 
per hour, while the sun apparently moved through 1 
sextant 5° 18’; how long did she sail, and how many 
statute miles? 


9. Washington is in longitude 77° 2’ 48” west, and 
Cincinnati is in longitude 84° 26’ west; when it is 6 a. 
m at either place what time is it at the other? 

10 If my watch keeps Terre Haute time and indi- 
cates 17 min. 10 sec. past 1 o’clock p. m. when it is 


& ms 


PROBLEMS. I23 


noon by local time, how far and in which direction am 
I from Terre Haute? 


11. A man travels from Pittsburg in longitude 80° 
2’ west, until his watch is 1 hour and 45 minutes too 
fast; how far and in which direction has he traveled? 
~12 The longitude of Galveston is 94° 50’ west while 
Constantinople is in longitude 28° 49’ east ; when it is 
noon at either place what time is it at the other? 

*13. A ship’s chronometer, set at Greenwich, indi- 
cates 5 hr. 45 min. 24 sec. p. m when the sunis on the 
meridian ; required the longitude of the vessel. 

’ 14 A degree of longitude in the latitude of Boston 


» - is 4 veoeraphic miles. How many more statute miles 
3 2g geograp 


in 7° of longitude at the equator than in the same 
number of degrees in the latitude of Boston? 

15. At 3 o’clock and 35 min. a. m. in London, what 
is the time at Boston? 

16. At 5 o’clock p. m.in Rome, what time is it in 
Terre Haute? 

17. At noon in Pekin what time is it in San Fran- 
cisco ? 

18. How many degrees of east or west longitude may 
a place have? Why? | 

19. What is the difference between the local and 
“standard” time of Terre Haute in 87° 20’ west ? 

20. When it is 10 o’clock at Indianapolis, what is 
“standard ” time at the same place? 


CHAPTER X. 
AREAS AND VOLUMES. 


130. Areas. 


Example 1. A room is 12 ft. long and 8 ft. wide ; 
how many square feet are in the floor? 


Solution. 
A surface 1 ft. l. and 1 ft. w.=1 sq. ft. 
A surface 12 ft. 1. and 1 ft. w.=12 times 1 sq. 
ft.=12 sq. ft 
A surface 12 ft. 1 and 8 ft. w.=8 times 12 sq. 
ft.=96 sq. ft. 
GLC 
Remark. Length, width and area are always expressed in 
concrete units, hence length cannot be multiplied by width; 
even if snch multiplication were possible, the product would 
not be area, which is different in name from the multiplicand, 
If, however, the numbers representing the dimensions of a 
parallelogram be multiplied together, the product is the number 
representing the square units in the given surface. 
Example 2. A lot contains 192 sq. rd. and.is 16 rd’ 
long; how wide is it? 


Solution. A surface 1 rd. 1. and 1 rd. wide=1 sq.rd. - 
A surface 16 rd. 1. and 1 rd. wide=16 times 
1 6g rd eae Oregera: 
A surface 16 rd. 1. must be as many rd. wide 
to contain 192 sq rd. as 192 sq. rd. are 
times 16 sq. rd., which—=12. 
*, the lot is 12 rd. wide. 


“ “Ny . aw 


* 


AREAS. 125 


LP omark. Tt the number of square units in a parallelogram be 
divided by the number of corresponding linear units in either 
dimension, the quotient is the number of linear units in the othcr 
dimension 

Exrample 3. The base of a plane triangle is six 
inches and its altitude is 4 inches; what is its area? 

Remarks. 1. The area of a plane triangle is one-half the 
area of a parailelogram having the same base and altitude as 
the triangle; hence find the area of such parallelogram as in 
Lx. 1, and divide it by 2. 

2. If the number of linear units in the base or altitude of a 
plane triangle be multiplied by one-half the number of like 
linear units in the other dimension, the product is the nwmber 
of corresponding square units in the triangle. 


Example 4. The diameter of a circle is 14 inches; 
what is its area? 

Remarks. 1. The circumference of a circle is nearly 34 times 
the diameter. : 

2. Any circle is practically equal to a rectangle whose length 
is the semi-circumference and whose width is the radius of the 
given circle. 

Solution. 84 times 14 inches=44 in.—the approxi- 
mate circumference of the given circle. 

Its equivalent rectangle is, therefore, 22 in. by 7 in. 
7 times 22—154,_ .*. the approximate area of the given 
circle is 154 sq. in. . 

Remarks. 1. The area of any circle is .7854 of the area of 
its circumscribed square. 

Fach side of the square that circumscribes the given circle is 
14 inches, the diameter of the circle. The area of the square 
is found by multiplying 14 by 14 and giving the denomination 
square inches to the product. © .7854 of this product equals 154-+ 
sq. in., the area of the given circle. 


2 The rule is usually formulated thus:—To find the area of a 
circle: Multiply the square of the diameter by .7854. 


126 VOLUMES, 


181. Volumes. 
Example 1. A common brick is 8 in. long, 4 in. 
wide and 2 in. thick; how many cu. in. does it contain ? 


Solution. 


A solid 1 in. 1. 1 in. w. 1 in. th.=1 cu. in. 
A solid 8 in. |. 1 in. w. Lin: th.=8 times 1 cu. in.=8 


cu. in. 

A solid 8 in. 1.4 in. w. lin. th.=4 times 8 cu. in.=32 
cu; In. 

A solid 8in. 1 4 in. w. 2 in. th.=2 times 32 cu. inj—— 
64 cu. in. } 


‘. a brick contains 64 cu. in. 


Remark. Tf the numbers representing the three dimensions 
of a rectangular solid be multiplied together, the product is the 
number representing the corresponding volume units in the 
given solid. 


4 


Example 2. A bin containing 315 cu. ft. is 9 ft. 
long and 5 ft. wide; how deep is it? 


Solution. 
A solid 1 ft. 1. 1 ft. w. 1 ft. d.=1 cu. ft. 


A solid 9 ft. 1. 1 ft. w. 1 ft. d.==9 times’ 1 cus it—9. 


cu. ft. 


A solid 9 ft. 1.5 ft. w. 1 ft. d.=5 times 9 cu. ft.—=45 
cu. ft. 


A solid 9 ft. 1. 5 ft. w. must be as many ft. deep to con- 


tain 315 cu. ft. as 315 cu. ft. are times 45 cu. ft. which 
equal 7... the bin is 7 ft. deep. 

Remark. Tf the number representing the volume of a rectan- 
gular solid be divided by the product of the numbers represent- 
ing twoof its dimensions, the quotient is the nwmber representing 
its third dimension. 


“Was 


PROBLEMS, 127 


Exercises. 


1. How many square feet in a surface 12 ft. by 5 
ft.? 
2. How many sq. yd. in a lot 15 yd. by 11 yd.? 
3. A field is 40 rods long and 18 rods Bite, how 
many acres in it? 
“4. A room is-15 ft. 6 in. long by 24 ft. 4 in. wide; 
how many square yards in the ceiling ? 2 
5. A floor contains 192 sq. yd., and is 12 yards wide; 
how long is it? 
6. A field contains 18 acres and is 60 rods long; how 
wide is it? 
7. A brick walk is # yd. wide; how long must it be 
to contain 25 square iaKdad 
8. A triangular wall is 6 ft long and 53 ft. high at 
its widest end; what is the area of one side? 
9. A block is 5 in. by 3 in. by 2 in.; what is its 
volume? 
10. A piece of stone is 7 ft. long, 53 ft. wide by 34 
ft. thick; how many cu. ft. in it? 
~11. Abin is8 ft. by 6 ft. by 45 ft.; how many bushels 
of wheat will it hold? 
12. A cistern is 9 ft. by 5 ft. by 5 ft.; how many gal- 
lons does it hold? 
13. How many-perch of masonry in a wall 350 ft. by 
18 ft. by 2 ft.? 
14 A block contains 64 cu. ft., and is 4 ft. wide and 
2 ft thick; how long is it? 
15. A box is 5 ft. long and 38 ft. wide ; how deep must 
it be to contain 60 cu. ft.? 
16. A tank is 10 ft. long and 4 ft. wide; how deer 
must it be to hold 3 tons of water? 
17. A havstack is 36 ft. long and 9 ft. wide; what is 
itx hight if it contain 11 tons of hay ? 
18. A horse is tied to a stake by a rope 100 ft. long 
over how much ground can he walk? 


CHAPTER XI. 
THE DECIMAL SYSTEM OF MEASURES. 


Remask. The history of the decimal system of measures and 
numerous facts and arguments showing its simplicity, are found 
in the publications of the American Metric Bureau of Boston. 

The purpose of this chapter is to exhibit the system and to 
present methods of computing by means of it. 


182. Primary Units. 
Of length, —the meter. marked m. 
“ capacity,— ‘‘ liter Cleeter.) “ 1, 
g. 


“¢ weight, — “ grim, : 


Remarks. 1. The ar is the unit for measuring land. Other . 


surfaces are measured by the sq. m. or by decimal parts or deci- 
mal multiples thereof. The ar is marked a. 

2. The ster is the unit for measuring wood. Other volumes 
are measured by the cubic meter or by decimal parts or decimal 
multiples thereof. The ster is marked s. 


3. In measuring great weights the quintal and the tonneau — 


(Metric ton) are used. These are marked Q. and T., respect- 
ively. | 


183. Secondary Units. 


Each of the secondary units used is either a decimal 
part or a decimal multiple of a primary unit, and is 
designated by using one of the following prefixes with 
the name of a primary unit. 


Decimal parts. Decimal Multiples. 
DécY¥, meaning .1. Déka. meaning 10. 
Céntt, of Ol. Hékto, 1S ee 
Milly, $ 001. KY16, 2 1000. 


Myria, é 10900. 


SCALES. 129 


Remarks. 1. In combining any one of the foregoing prefixes 
with the name of a primary unit, each word retains its own 
pronunciation and accent. 

2 In abbreviating a word formed by combining a prefix with 
the name of a primary unit, it is customary to use the initial 
letter of each word, using a small letter to designate the primary 
unit or any of its decimal parts, and a capital letter to designate 
any of its decimal multiples. 


Scales. 


184. Of Length Measure. 


10 10 eee 10 10 10 10 10 
Rem eteriee bim,. Dm. ©m: dm: cm. mm, 
1 1 iF 1 1 1 if i 


185. Of Surface Measure. 


100 1003¢ =100 100 100 100 100 
Sq. Mm sq. Km. sq. Hm.sq. Dm. sq, m sq.dm, sq. cm,sqe mm, 
1 1 1 1 1 Leek A 1 


186. Of Land Measure. 


: Meme 10% a a0, 10...) 10 
Mav iKa, ia: . Da. Mimic aon a Ge. vue TE 
1 1 1 1 1 1 h: 1 
2 


187. Of Volume Measure. 


1000 ~=1000 1000 1000 1000 1000 1000” 
Cu. Mm. cu. Km. cu. Hm. cu. Dm.cu. m.cu. Dm. cu. em cu. mm. 


1 1 1 1 1 1 1 


188. Of Wood Measure. 


Remarks. Only three units are used. 
10 10 
Der s.. =ds. 
go ei | 1 


ee 


120 METRIC SYSTEM. 


189. Of Liquid and Dry Measure. 


10°) 10°)"'10*. 10" 10 ee 
M1. Kl. HL. Die. cL > dip eee 
1 1 1 1 ] 1 i 1 


190. Of Weight. 


10 10 10 10 10 10 10 10 10 
TT.’ Q., Mg... Kg. Hg. De?~ peo doeGeeeeeen 
1 see 1 1 1 a) 1 1 1 


Relation of Decimal and Common Measures. 


191. Of Length. 192. Of Surface, 
1 cm.=.3937 in. 1-sq. dm.=154 sq in. 
1 dm.-=3.937 in. 1 sq. m. =1550 sq. in 
1 m=39'57 90. {pee ! 119.6 sq. yd. 
1 Dm.==32!8 Tt; 1 sq. Dm. 
1 Hm,.=828 ft. 1 in. 1 Ha.=2.47 acres. 
1 Km.=8280 ft. 10 in.=% mi., nearly. : 
1 Mm. 6.2137 miles. 


193. Of Vora: 


1 cu. em’ —.06 cu. in. 
1 cu. dm.=61 eu. in. 


Leu: on) 39.317 cu. ft., 
or -.= or 
1 ster. J 1,308 cu. yd. 
194. Of Capacity. 


1ml.==1' curiem, =327 fi dr.== 061m 

1c). 10 cu. cmiss338 f1.'07 == G17 ae 

1 dl.=.1 cu: dm. 845 91 261. SEAR, 

1 1.=1 cu. dm.=1605 qt. lig.==.9 qt. dry: 

1 D1 =10 cu. dm.=—2.64 gal. liq.==9.08 qt. dry. 
1 Hl.=,t.cu. m.=26.41 gal. lig.=2:84 bu. 
1.K1.=1 cu. m.=264-17 gal. liqg-—1 30S cue 


RELATIONS. 


195. Of Weight. 


131 


1 mg.=the wt. of 1 cu mm. of water=.015 gr. Troy. 
errs eg LO i es TOE vt 
NE ENGIN Fb oi lois tte iah Mie Bese =e 
a SUMMERS BCMA UA etadn— s 
Me ea 1 eta) EN Aa 8h Om 
Pee nl dhe Oe. hes 3.53 
Meo i LS SP a fobs pene PPD 

Be ect 6 10. ie near 2a Jats 

ie aon oe aE val nep ead DU At? Ss 

Pee ee eum. eM . §© == 9204.6; * 


Remark. 


The fractions in the above tables of relative value 


are not exact, but they are sufficiently approximate to meet 
most applications. Indeed where no great accuracy is required 
it is sufficient to use the following table of— 


196. Approximate Values. 


1 dm. 


sq. m. == 
Ha. = 


cu. m, or ster== 


el gee coe ee oe oe 
— 
‘ 
l| 


ce 


C6 


=-about 4 in. 


o9F In, 

1 rod. 

= mi. 

10? sq. ft. 
24 acres. 
13 cu. yd., or } cord. 
1 qt. 

24 bu. 

153 gr. 

24 |b. 

ae 2200-1b. 


‘ et = § Fe 1 Pe OMe 
1, Sai 


132 . MEPRIC SYSTEM, 


197. Table of Specific Gravities, Water Being I. 


Air, j 001 Ice, 93 
Alcohol, pure, wo Iron, cast, oe 

‘ commercial, .83 ‘““ wrought, ie 
Brass, pee) Lead, i es 
Brine, 1.04 Lime, 1.8 
Coal, soft, 1.25. Marble, 2.7 

“hard, ind Mercury, 13.6 
Copper. 9. Milk, 1.03 
Cork, “4, Silver, 10.5 
Gold, 19.2 Zine. | tel 
198. Important Facts. ° 


1. The ar is an area equivalent to 1 sq. Dm. 
2. The ster (pronounced stair) is a volume equiva- 
lent to 1 cu. m. ; 


9 


3. Thgeliter is a volume equivalent to 1 cu. dm. 


4. The gram is the weight of a cu. cm. of distilled - 


- . . . . 
water at its greatest density; 1. e., at 4° centigrade, or 
39.2° Fahrenheit. The water is balanced in a vacuutnh. 


5. f liter, or 1 cu. dm. of distilled water weighs 1 _ 


kilogram, while 1 cu. m. of water weighs 1 metric ton. 


6. The legal letter weight in the United States is 


1 oz. Av. The Postmaster General is authorized to 
substitute a 30 gram weight for the 1 oz. weight in all 


postoffices that exchange mails with foreign nations ~ 


and in other postoffices at his discretion, 
7. <A freshly coined nickel 5 cent piece (not the 
““V’’ nickel), is 2 cm. in diameter and weighs 5 grams. 
The silver coinage of the United States is worth 4 cents 
per gram: Six 5 cent nickels orone hundred twenty 


cents in silver coin constitute one letter weight. 


PROBLEMS, E34 


Exercises. 

Remarks, 1. Areas and volumes are found as in the common 
measures. 

2. In finding the capacity of a given volume it is necessary 
to remember that a cu. dm = 1 liter. 

3. In finding the weight of a given volume or capacity of a 
substance, it is necessary to remember that a cu. cm of dis- 
tilled water weighs 1 g., that 1 cu. dm. of distilled water weighs 
1 Kg. and fills a liter cup, and that 1 cu. m. of distilled water 
weighs one metric ton. If the weight of a substance other 
than water be required, we find the weight of an equal vol- 
ume of water and multiply it by the specific gravity of the 
substance. . 

_ Example1. A tank is 4m. long, 3 m. wide, and 2.5 » 
m. deep; how many Keg. of brine will fill it ? 

Solution. 4X38X2.5=30=the number of cu. m. in 
the tank. 

Since 1 cu. m. of distilled water weighs 1 metric ton, 
30 cu. m. of distilled water weigh 30 metric tons; and 
‘since 1 T.-=1000 Kg., 30 T.=380000 Kg.—the weight 
of the water necessary to fill the tank. 

Since brine is 1.04 times as heavy as water; the 
weight of this volume of brine = 1.04 times 30000 Kg., 
Ona 2s ge. ..°, otc. 
~2. Find the area of the walls of a room 6.2 m. long, 
- 5.05 m. wide and 3.5 m. high. 

8, How many rolls of paper 45cm. wide and 6.2 m. 
long, allowing 11.2 sq. m. for openings will be required 
to paper those walls ? 


4, Find the cost of plastering the above room at 
50¢ per sq. m. 

5. How many sq. m. ina board 8m. by 25 cm.?, 5 
m. by 25cm.? 7 dm. by 156 mm,’ 


6. If wood is cut into 120 cm. lengths, and a pile is 


134 METRIC SYSTEM. 


43.7 m. long and 1.6 m. high, how many sters does it 
contain ? 


7. A bin is 11.4 m. long, 4.15 m. wide, and 2.8 m. 
deep; how many hektoliters of barley does it hold ? 

8. If the specific gravity of grain is .81, what is the 
weight of the grain that fills the bin ? 

9. A vat is 186 cm. long, 7.7 m. wide and 48 dm. 
deep; how many tonneaus of water does it hold? 

10. What is the weight in kilograms, of a cu. cm of 
water? Ofa Hl. of water? Of Mercury? Of milk? 
Of lime? Ofacu. m. of cork? Of brass? Of zine ? 

11. What weight of water will fill a vat 92 cm. by 
76cm. by 4.2 dm.?) What weight of milk will fill it? 

12. If the above vat be filled with brine weighing 
1.01 Kg. per liter, required the weight of the brine. 

13. How many liters of air in a room 6.3m. by 6.17 
m. by 2.9m ? What is the weight of the airin grams? 
In kilograms? 

14. How many pills of .36 g. each, can be made from 
a mass weighing .72 Kg. ? 

15. What is the weight of 7.1 HI. of pure alcohol? 

16. An irregularly shaped mass of copper displaces 
.88 1. of water ; what is its weightin kilograms? 

17. <A piece of iron 125 cm. long, 56 cm. wide. and 


6.2 cm. thick weighs 256.4 kilograms ; what is the spe- 


cific gravity of the iron? 
18. A piece of ore weighing 5.6 kilograms, weighs in 
water only 3.12 kilograms ; what is its s. g.? 
19. Ifa tap running 2.71. per min, fill atubin 14 
minutes, how long would it take a tap running 4.2 1. 
per minute to fill it? 


PROBLEMS. 135 


20. A cistern will hold 18 tonneaus of rain water ; 
what depth of rain must fall upon a flat root 20 m. 
long, by 15 m. wide, to fill the cistern ? 


21. Find the weight in Kg. of a block of ice 4.5 m. 
by 3.2 m. by 2 dm. 


22. What is the weight of a bar of lead 2.3 dm. by 2 
em. by 1.2 cm.? 

23. What is the weight of a piece of copper 4 dm. by 
16dm. by 3 cm.? 

24. What is the value of a piece of silver 3.2 dm. by 
Tcm. by 2 cm. at 3¢ per gram? 

25. How many bullets, each weighing 2.7 deka- 
grams can be made from a cubical block of lead whose 
edge is .74 dm. ? 

26. What is the s. g. of a substance that weighs 20 
Kg. in air and 18 Kg. in water? 
| 27. How many jets, each burning 110 liters of gas 
per hour for 23 hours each night, would consume the 
contents of a gasometer containing 300000 cu. m. of 
gas? 
| 28 What is the weight of a block of marble .72 m. 
_'by 12 dm. by 1.2 dm.? 

29. A box is 2.3 m. by 9dm. by 25cm.; how many 
liters does it hold? 


. 30. If a wheel is 90cm. in circumference, how many 
times does it revolve in going 5 kilometers ? 

31. If acu. cm, of ore weigh 6.2 g; required its s. g. 
_ 32 How many meters of carpeting .7 m. wide musi I 
buy to cover a floor 6 m. by 7 m., the strips extendirg 
the longer way of the floor? 

33. Required the area of a circle 3.2 m. in diameter. 


136 METRIC SYSTEM. 


34. If acu. m. of earth weigh 1268 Kg., what is its 
s. g.? 

35. A cistern is 4 m. deep by 3 m. wide, how long 
must it be to hold 60 metric tons of water ? 

36. What is the Troy weight of 3 cu. dm. of mer- 
cury ? ; 

37. How many cu. cm. in a stone whose s, g. is 2, 
and whose weight is 5.74 Hg.? 

38. What is the weight in Kg. of a substance whose 
volume is 9.4 cu. dm. and whose s. g. is .875? 

39. Three liters of oil weigh 2.769 Kg.; required its 
_ 8. 8: 

40. A church bell in Montreal weighs 11263.6 Kg. 
If its s. g. is 8.7, how many cu. dm. of metal does it 
contain ? 

41. A piece of mefal immersed in a vessel full of 
water, caused 1.374 1. to overflow ; w hat was the weight 
in Kir of the metal? 

42. If 8 cu, m. of sand weigh 7. 134 T.; 3 its 
Ss, 

re A box is 1.17 m. by .9 m. by 1.04 m. inside meas- 
ure; how many bars of soap 13 cm. sq. by 29 em. I. »g 
apuld be packed in it; .12 of the capacity being de- 
ducted for packing. | 

44. If a ster of cork is worth $20; what is the value 
of 100 Kg. of cork—weighing } as.much as water? 

45. Sea-water contains 28 parts, by weight, of salt in 
1000. A liter of sea-water weighs 1.025 Kg. How many 
Kg. of salt can be obtained from7126.276842 cu. m. 
of seawater ? 

46. A book is 2.1 ecm. thick; each leaf is -05 mm. 
toick ; how many pages in the book ? ae 


4 


GHAPTER XII. 
PERCENTAGE. 


199. Percentage is a method of computing by hun- 
dredths. 


The Terms Employed. 


200. The Base. The base is the number of which a 
number of huadredths is reckoned. 


20!. The Rate Per Cent. The rate per cent. is the 
fraction which indicates the number of hundredths in- 
volved 


202. The Percentage. The percentage is a number 
that bears the same relation to the base that the rate 
G does to 1. Itis a number of times .01 of the base 
if the rate % is .O1 or more; it is a part of .01 of the 
base if the rate % is less than .O1. 


Remarks. 1. The term amount is applied to the sum of the ; 
base and the percentage. 


The base is +?4 of itself and the percentage is a number of 
hundredths of the base, hence their sum, the amount, is annm- 
_ ber of hundredths of the base, and comes within a definition of 
the percentage. 

The term difference is applied to the part of the base that 
remains after the withdrawal of the percentage from the base, 
The difference is, therefore, a number of hundredths of the base 
and comes within the definition of the percentage. 


10 


138 PERCENTAGE 


Relations of Percentage. 


203. To Muitiplication. The percentage, being anum- 


ber of hundredths of the base, is found by obtaining. 


the number of hundredths -of the base that the rate 
per cent. indicates; i. e., by multiplying the base by 
the rate per cent. The base and the rate per cent. are 
thus seen to be factors of the percentage. 


Percentage is related to multiplication in the signifi- 
cation of its terms; the base being multiplicand, the 


rate per cent. being multiplier and the percentage being 


product. 


204. To Factoring. Principle 1, If two or more fac- 
tors are given, their product is found by multiplying 
the factors together. 


Principle 2. If the product of two factors and one 
of them be given, the other is found by See the. 


product by the given factor. 


In the light of one or the other of the above princi- 
ples of factoring is seen the process to be performed in 
finding any one of the three terms of percentage if the 
other two are known. 

Percentage is related to factoring in the principles 
which determine the processes to be performed. 


205. ToFractions. The rate per cent.is always given 
in the denomination of hundredths. 


Percentage is related to fractions in that one of its 
terms (the rate per cent ) is a fraction. 


a 


SOLUTIONS. 139 


General Cages of Percentage. 


Remark. All special problems in percentage may be classified 
under three cases. 
206. Case I. Given the base and the rate per cent. 
to find the percentage. | 


Solution. Since the percentage is the product of the 
base and the rate per cent., the percentage is found by 
multiplying the base by the rate per cent. 

207. Case Il. Given the base and the percentage to 
find the rate per cent. 

Solution. Since the percentage is the product of the 
base and the rate per cent, the rate per cent. is found 
by dividing the percentage by the base, expressing the 


- quotient.in the denomination of hundredths. 


208, Case Ill. Given the percentage and the rate 
per cent, to find the base. 

Solution. Since the percentage is the product of the 
base and the rate per cent., the base is found by divid- 
‘ing the percentage by the rate per cent. 

Remark. It is observed that Case I issolved by multiplica- 
tion in the light of the first of the two principles stated under 
the second relation of percentage, while Cases II and III are 
‘both solved by division in accordance with the second of the 
two principles referred to. On the basis of processes employed, 
two cases will be found to embrace all problems in percentage. 


209. Forms of Solution. 
Example 1. Whatis 7 % of 385? 
First form. 
We have given the base, 85, and the rate per cent. 
.O7, to find the percentage. Since the percentage is the 


_ product of the base and the rate per cent., the per- 


centage in this example, is found by multiplying 35 
by .07. The product, 2.45, is the required percentage. 


140 PERCENTAGE. 


Second form. 
1% of 35:=.01 of 835=.35. 
7 % of 35==7 times .85 % 2.45. 
7 % of 35=2.45. 
Example 2. 4.5=what % of 75? 
First form. 

We have given the base, 75, and the percentage, 4.5, 
to find the rate per cent. Since the percentage is the 
product of the base and the rate per cent., the rate per 
cent., in this example, is found by dividing 4.5 by 75, 
expressing the quotient as hundredths. The quotient, 
06, is the required rate per cent. 


Second form. 

1 % of 75=.01 of 75=:.75; hence 4.5 equal as many 
times 1 % of 75 as .75 is COTATI AS times in 4.5 which 
=—6. 6 times 1 %=—6 &. es 

Third ee 

* L=-, of 75, 45 =45 times ~. of 75 = 42 of 75. 
Multiplying both terms of #2 by 14, we have .06. 
45=6 2% of 75. 


Example 3. What per cent. of a number is 35 
of it? 


ie Sh oe 7 2s 1?) 
ois =o; Hence 3 of a number ==-.30, or 30,7 


Example 4. 36 equals 9% of what number? 
First form. 

We have given the percentage, 386, and the rate per 
cent., .09, to find the base. Since the percentage is the 
product of the base and the rate per cent., the base in 
this example, is found by dividing 36 es 09. The 
quotient, 400, is the required base. 


SOLUTIONS. | 141 


Example 5. 48—=20% more than what number ‘ 
First form. 

Since 48 =20 % more than some number, 48 = 120 
% of that number. We now have given the percent- 
age, 48, and the rate per cent, 1.20, to find the base. 
Since the percentage is the product of the base and the 
rate per cent., the base in this example, is found by 
dividing 48 by 1.20. The quotient, 40, is the required 
number. 

Second form. 

Since 48 = 20 % more than some number, 48 = 120 
% of that number; and 1 % of the number must = 
iy Of 48 =.4, and 100 % of the number must = 100 
times .4—40. .. 48—20% more than 40. | 

Third form. 

Since 20% of anumber =} of it, then 48 is $ of 
some number. 

4 of that number must = #4 of 48 = 8; and 
= of the number — 5 times 8 = 40. 
*, 48 = 20% more than 40. 


Example 6. %==10% lessthan what number? 


. First form. 
Since 3 = 10% less than some number, = 90% of 
that arabe. 


We now have given the percentage, 3, and the rate 
per cent., .90, to find the base. 

Since the percentage is the product of the base and 
the rate per cent.. the base in this example 1 is found 
by dividing 2 by .90. the quotient, %, is the base. 


Second form. 
Since 10% less than some number = 3, 
90% of the number must =3; 
and T % 66 66 66 7S — aa of 3—= 145, 
6c 100% 66 66 66 66 =100 times apiece’ 
“.8=10% less than 4. 


PERCENTAGE, 


. Exercises. 
icant Eee 

% ; Aad % 

9 (75 

aes 


ne Oa 
2 


= 


e . sa ow bo 


° 


5 
6 
< 
8. 
9 
iO 


PROFIT AND LOSS. 143 


Applications of Percentage. 


210. There are two classes of applications of per- 
centage. The first class includes all those problems in 
which the percentage is the product of the base and 
the rate per cent. 

The second class includes those problems in which 
the percentage is the product of three factors, viz., the 
base, the rate per cent., and the number representing 
the time (in years) involved in the transaction under 
consideration. } 


2i!. The principal applications of the first class are 
— Profit and Loss, Commission and Brokerage, Stocks, 
Insurance, Taxes and Customs. 


The-principal applications of the second class are— 
Interest, Discount, Bonds, Exchange, Equation of 
Payments and Accounts. 

Remarks. 1. In every problem of the first class the three 
terms of percentage are represented. 

2. Many of the problems to be solved are compound, being 
composed of two or more problems, some of which may not in- 
volve percentage. 


Applications of the First Class. 


212. Profit and Loss. 
CORRESPONDING TERMS. | 

(1.) The nun ber on which the gain or loss is es- 
timated is the base. In most examples the cost price 
is the base. 

(2.) The gain, loss or selling price is the per- 
centaqde, 

(3.) The rate of gain, rate of loss or rate of se...ng 
is rate per cent, | 


144 _ PERCENTAGE. 


213. Forms of Solution. : 
Example 1. A man paid $110 for a horse and sold 
it at a profit of 20 per cent.; required the gain. 


First form. 


We have given the cost, $110, which is base, and the 
rate of gain, .20, which is rate per cent., to find the gain, 
which is the percentage. 

Since the percentage is the product of the base and 
the rate per cent., the percentage in this example, is | 
found by multiplying $110 by .20. The product, $22, 
is the required percentage, which is the gain. 


Second form. 


1% of $110=— .01 of $110 = $1.10, and 
20% of $110= 20 times $1.10 = $22. 
.. The gain was $22. 


Third form. 
20% of a number is 4 of it, 
20% of $110 =1 of 110 = $22. 
.. The gain was $22. 


Example 2. Ifamerchant pay 15 cents per yard 
for muslin, for how much does he sell it to lose 25% ? 


First form. 


‘Since he sells at 25% below cost, he sells for 75% of 
cost; and we have given the cost price, 15¢, which is 
] ase, and the rate of selling, .75, which is rate per cent, 
to find the selling price, whichis the percentage. Since 
the percentage is the product of the base and the rate 
per cent., the percentage in this example is found by 
multiplying 15¢ by .75; the product, 11}¢, is the re- 
Guired percentage, which is the selling price. 


SOLUTIONS. 145 


Second form. 


Since he sells at a loss of 25%, he sells for 75%, or ? 
of the cost. 


Sof 15¢ = 3times 43¢ = 11} ¢. 
-*. Otc. 


Example 3. <A hat costing $8 was sold for $9, what 
was the rate of gain? 


First form. 

(1.) $9, the selling price, minus $8, the cost, =$1. 
the gain. 

(2.) We now have given the cost, $8, which is 
base,and the gain, $1, which is the percentage, to find 
the rate of gain, which is rate per cent. 

‘Since the percentage is the product of the base and 
the rate per cent., the rate per cent., in this example, 
is found by dividing $1 by $8, expressing the quotient 
as hundredths. The quotient, .124, is the required 
rate per cent. which isthe rate of gain. .°. etc. 


Second form. 
$9, the selling price, minus $8, the 
cost price, = $1, the gain. 
$1 =+ of §8. 
+ of a number —.124 of it. 
.. the rate of gain was 124%. 


Example 4. <A grocer sold coffee at 8¢, above cost, 
and gained 20 per cent.; required the cost. 
Form. 

We have given the gain, 8¢, which is the percentage, 
and the rate of gain, .20, which is rate per cent., to 
find the cost, which is base. 

Since the percentage is the product of the base and 


146 PERCENTAGE 


the rate per cent., the base, in this example, is found 
by dividing 8¢ by .20; the quotient, 40¢, is the re- 
quired base which is the cost. .*. ete. 


Example 5. A merchant marked goods at 20% 


above cost, and then sold at 20% less than the marked 
price. Did he gain or lose and how much per cent. ? 


Form. | 
(1.) Assume $1 as the cost. 


(2.) Since he marked 'the goods at 20% above cost, 
the marked price was 1.20 of. the cost; and we have 


given the cost, $1, which is base, and the rate of mark- 


ing, 1.20, which is rate per cent, to find the marked 


price which is percentage. [Solving by Case I, the’ 


marked price is found to be $1.20.] 


(8.) Since he sold at 20% less than the marked 
price, he sold for .€0 of the marked price, and we have 
given the marked price, $1.20, which is base, and the 
rate of selling, .80 which is rate per cent, to find the 
selling price which is the percentage. 

Since the percentage is the product of the base and 
the rate per cent., the percentage in this example, is 
found by multiplying $1.20 by .80, the product, $.96 
is the required percentage, which is the selling price. 


(4.) The cost, $1, minus the selling price, $.96 = 
$.04 = the loss. 


(5.) We now have given the cost, $1, which is 


base, and the loss, $.04. which is percentage, to find the 
rate of loss, which is rate per cent. This isfound (Case 
II) by dividing $.04 by #1 andexpressing the quotient 


as hundredths. The quotient, .04, is the required rate 


per cent., which is rate of loss. 


PROBLEMS, Der ae. 147 


Exercises. 


1. 14 barrels of flour are bought at $3.874 each and 
sold at $4.124 each ; peace the gain. The rate of 
gain. © © 

2. A horse is Bedeht for $94, and sold at 4% gain ; 
required the gain. The selling price. » 


3. A lot is bought for $2256; for what must it be 
sold to gain 20 per,cent.? 


4. There was a gain of $14 realized on a certain 
sale; the rate of gain was .15; what was we PPT 
price? The selling price? ‘33° 


5. A grocer bought 148 gallons of molasses at 26¢ 
per gal., and sold it for $64; did he gain or lose? Bow 
much and at what rate ? 

6. A farm was sold for $7400 which was 5% more 
than it was worth; required its value. 

7. A man sold two horses for $120 each ; on one he 
gained 15% and on the other he lost 15% ; did he gain 
or lose on the entire operation, and how much ? 

8. I sold a piece of property for $1000 gaining 16%; 
I then invested the $1000 in another piece of property 
which I was forced’ to sell at a loss of 16% ; did I gain 
or lose on the series of transactions and at what rate? 

9. A quantity of wheat was sold for $1248 which 
was 12% more than its cost ; required the cost. 

10, If the cost and rate of gain are known, how find 
the gain? The selling price. Form and solve a 
problem. 

11. If the gain and the rate of gain or loss be given 
what can be found and how? Form and solve a 


problem. 


148 PERCENTAGE. | 


12. If cost and selling price are known what can be 
found and how? Form and solve a problem 


13 A man bought corn at 30” per bu. and sold it at 
a gain of 163% ; what was the selling price? 

14. If land cost $48 per acre, how much must be 
asked for it, that a 10% abatement may be made anda 
profit of 14% still be realized ? 

15. A merchant asked an advance of 35%, but after- 
ward sold ati25% less than his asked price; did he gain 
or lose and how much on goods that cost 272.25? 

16. I sold ? ofan article for what the entire article 
cost; what was my rate of gain? 


17. A grocer sold a hogshead of molasscs for $30 
which was 15% more than it cost; required the cost 
per gallon. 


18. A merchant marks a piece of goods $12, but 
takes off 6% for cash. If he still makes 10% profit, 
what was the cost of the goods? 

19. A grocer gains 14% by using a false weight 
when selling goods ; required the weight of his pound 
weight. ! 

20. A man bought a horse for $55 and sold him for 
$70; required his rate of gain. 

21 If#of an article be sold for what % of its cost, 
what is the rate of gain? A 

22. If butter be bought at $24 per cwt., for what 
price per lb. mustit be marked to gain 15%, and allow 
a discount of 8% for cash? 

23. A merchant bought gloves at $15 a doz.; how 
must he sell them by the pair, to gain 20% ? 


24. If a house be bought for $3850 how should it be 
sold to make 123% on the purchase? 


COMMISSION. —_- 149 


25 Ifa milkman sells 8 quarts of milk for the price 
he -pays for one gallon, what per cent. does he make? 

26 <A grocer makes 3¢ on every egg he sells, which is 
25 Y% profit, what do they cost him per doz.and how 
does he sell them per doz.? 

27 I bought wood at $44 a cord, and sold it for $6}; 
required the per cent. profit. 

28. A dealer bought 1650 tons of ice, at $12 a ton, 
one-half of it melted, and he sold the remainder at $1 
per hundred ; what per cent. was the loss? 

29. I pay $2 for 3 lbs. of tea, and sell 2 Ibs. for $3, 
what is the per cent. profit? 

80. A stove dealer paid $30 apiece for stoves; how 
shall he se!l them that he may make 20% of the cost? 

31. dfa—bookseHer—-marks—his-goods-at 25% above 

~cost,-What-per-cent:-does-he make-or lose? 
32. Do I make or lose in selling an article marked 
5% above cost, if I deduct 20% ? 

33. What must be the Sos pee and profit of 
coal whose first cost is $6. 

84. I sold goods, Sanka 40% above cost, at a deduc- 
tion of 835%. What per cent did I lose? 


215. Commission and Brokerage. 
CORRESPONDING TERMS, 

1. The sum representing the amount of business 
transacted is base. 

2. The commission or brokerage is the percentage. 
The amount involved, or the sum of the investment 
and the commission is the percentage in some Instances. . 

3. The rate of commission or brokerage 1s rate er 
cent. If the sum of the investment and commission 


150 PERCENTAGE. 


be the percentage, the rate per cent. is 19% + the rate 
of commission. 


216. Forms of Solution. 


Example 1. An agent sold $1560 worth of goods 
and charged 4% for his services ; required his commis- 
sion. 


Form. 


We have given the sum representing the amount of 
business done, $1560, which is base, and the rate of 
commission, .04, which is rate per cent., to find the 
commission, which is the percentage. Since the per- 
centage is the product of the base and the rate per 
cent , the pe centage in this example is found by mul- 
tiplying $1560 by .04. The product, $62.40, is the per- 
centage, which is the required commission. 

[Give other forms of solution]. 


Example 2. An agent was paid $24 for buying 
$1440 worth of wheat; required his rate of commis- 
sion. 


Form. 


We have given the commission, $24, which is per- 
centage, and the sum representing the amount of bus- 
iness done, $1440, which is base, to find the rate of 
commission which is rate per cent. . 

Since the percentage is the product of the base and 
the rate per cent:, the rate per cent., in this example is 
found by dividing $24 by $1440, making the quotient 
hundredths The quotient, .01#, is the rate per cent. 
wich is the required rate of commission. .*. ete. 

[Give other forms of solution. ] 


- SOLUTIONS. } 151 


Example 3. A banker received $40 for making a 
collection at 4% ; required the sum collected: 
Form. 

We have given the commission, $40, which is per- 
ecntage, and the rate of commission, .04, which is rate 
per cent.,.to find the sum representing the amount of 
business done, which is base. 


Since the percentage is the product of the base and 
the rate per cent., the base in this example is found by 
dividing $40 by .04. The quotient, $1000, is the base, 
which is the sum representing the amount of business 
done. .*. etc. 


[Give other forms of solution. ] 


Example 4, A commission merchant received $4160 
with which to purchase goods after deducting his com- 
mission of 4% ; required the sum invested. 

First form. 

Since the rate of commission was .04, the sum received 
wos 1.04 of the sum invested; and we have given the 
amount received, $4160, which is percentage, and the 
relation of this sum to the sum invested, 1.04, which 
is rate per cent., to find the sum invested which is base. 

Since the percentage is the product of the base and 
the rate per cent., the base in this example is found by 
dividing #4160 by 1,04. The quotient, $4000, is the 
base which is the required sum invested. 

Second form. — 

Every dollar’s worth of goods purchased cost the 
sender of the money $1 for the goods +4¢ for agent’s 
commission, or $1.04; hence as many dollars wer: 
spent for goods as $1.04 are contained times in $4160, 
or 4000. .. the sum spent for goods was $4000. 


OE I et ee a eae ae 
ve. » SRN ~ Pe ay Hs Ad, 


152 PERCENTAGE, 


Third form. 
Since the sum invested + 4% of itself = $4160, 104% 
of the sum invested = $4160, 
and 1% of the sum invested = +}, of $4160 = $40, 
BAUM pin aw eee ‘¢ = 100 times $40 = $4000. 
.. The investment was $4000. 


Exercises. 


* 1. At 3% commission what does an agent receive 
for selling $15360 worth of goods? 

2. A lawyer collected 65% of a debt of $348; his” 
rate of commission was 4% ; what did he receive ? 

o. A merchant sent his agent $1250 with instruct- 
tions to buy goods ; how much was expended for goods 
after the agent deducted his commission at 25% ? 

4. How many barrels of flour at $4.50 per barrel 
can be bought for $684, if 3% commission be deducted 
before purchasing ? 

5. An agent received #240 as commission at 4% ; 
what amount of business did he transact ? 

6. The rate of commission was 5%; the sum sent 
the owner as proceeds of the sale was $2275; what was 
the commission ? 

7. What amount of business is done if $36.50 is the 
commission at 23% ? 

8 I sent my agent $516 to invest in corn after de- 
ducting his commission at 35% ; what sum was paid for 
corn ? 

9. An agentreceives the sum of $758 with which 
to buy goods, deducting his commission at 17% ; what 
is his commission ? 

~—“t0. If the merchant for whom the business is done 


‘ate 


BA Mae ee 
ay see 


PROBLEMS. 153 


(in Ex. 9) send check for the commission, what is its 
face? 


11. A consignment of grain was sold for $15650, of 
which $14550 were net proceeds; required the rate of 
commission. 


12. An agent received 2% commission for asnine 
goods. His entire commission was #126; what sum 
did he remit to his employer ?, 

13. I sold goods on commission at 4% through a 
broker who charged me 23%; my commission after 
paying the broker was $216; required the net proceeds 
of the sales. 


_~14. A real estate agent retains as commission $124, 


sending the employer $6575; what was the amount of 
the sale and the rate of cominission? 

15. An agent bought 25 horses on commission at 
34%. His commission was $63; required the cost of 
each horse. 

16. If I pay an attorney $48.50 for making a collec- 
tion at 5% ; what was the claim? 

17. How many barrels of flour at $5 each can an 
agent buy for $324, deducting his commission of 3% ? 

J8. An agent received $3440 with which to buy pork 
after deducting his commission of 1}% ; required the 
number of pounds of pork purchased at 3¢ per pound? 

19. A shipment of hay sold for $14 per ton; the rate 
of commission was 3%; incidental charges $200; the 
net proceeds were $6290. Required the number of 
tons. 

20. An agent received $5650 to invest in corn; after 
deducting his commission of 33 per cent., how much 
was invested in corn, and what was the agent’s com- 
mission ? 

11 


154 PEKCEN PAGE, 


21. An agent received $13000 with which to buy 
horses; after deducting’ his commission of 4} per cent., 
how many dollars did he spend for horses ? 

22. An agent sold a quantity of metals, and after — 
retaining 34% commission he sent the consignor $15246; 
required the amount of the sale and the commission. 

23. A commission merchant having sold some goods, 
paid $4168.80 to his employer, and retained as his com- 
mission $151.20. What was the rate of his commis- 
sion ? 

24. I remit to an. agent $360.70 with which to pur- 
chase goods; after deducting his commission of o% and 
paying $3.70 insurance what sum does he invest in the 
goods ? 

25. A commission Pere in Burlington was or- 
dered to buy flour at $5.75 a barrel, and received for 
that purpose $5063.50 which was to cover both the pur- 
chase and the commission at 4%. How many barrels 
of flour were bought ? 

26. An agent remitted to the consignor #9384, as 
net proceeds of a sale for which he received 3% com- 
mission. What was the sale? 


217. Stocks. 
CORRESPONDING TERMS, 

1. The par value of stocks is base. Sometimes the 
conditions are such that the cost or another number 
becomes base. 

2. The premium, the discount, or the cost, is ee 
pile is 

. The rate of premium, rate of discount, rate of 
Ree or rate of sale is rate per cent. 


- SOLUTIONS. 155 


18. Forms of Solution. 


Example 1. A company pays a dividend of 5%; 
what does that man receive who owns 12 shares? 


Form. 


12 shares of $100 each are worth $1200 We now 
have given the par value of the stock, $1200, which is 
‘base, and the rate of premium, .05, which is rate per 
cent., to find the premium, which is percentage. 
Since the percentage is the product of the base and 
‘the rate per cent., the percentage (in this example) is 
found by multiplying $1200 by .05. The product, $60, 
is the percentage, which is the required dividend. 
[Give other forms. ] 


Example 2. How much bank stock at 10% dis- 
count, can be bought for $27945 ? 


First form. 

At 10% discount, the stock is bought at 90% of the 
par value; and we have given the purchase price, 
$27945, which is percentage, and the rate of purchase, 
.90, which is rate per cent., to find the par value, which 
is base. 

Since the percentage is the product of the. base and 
the rate per cent , the base in this example is found by 
dividing $27945 by .90. The quotient, $31050, is the 
base, which is the required par value. 


Second form. 

Since the stock is bought at 10% discount, every $1 
‘worth of stock is bought for $.90.. As many dollars’ 
worth of stock can be bought for $27945 as $.90 is con- 
tained times in $27945, which equal 81050, _ .*. $31050 
‘worth of stock can be bought for #27945, at 10% dis- 
count. bho g) . 


TAY Ry Me kg ee ee eat, so | 
« eae i. pore ~* ke " ¢ yen 
f Beery | 


156 PERCENTAGE 


Example 8. Stock was bought at 120 and sold at 
128; what was the rate of gain ? 


Form. 


At 120, $1 worth of stock was bought for $1.20, and 
at 128 it was sold for $1.28. The gain was %.08. We 
have given the cost of the stock, $1.20, which in this 
case is base, and the gain, $.08, which is percentage. 
Since the percentage is the product of the base and the 
rate per cent., the rate per cent. in this example is found 
by dividing $.08 by $1.20, expressing the quotient as 
hundredths. The quotient, .06% is the rate per cent., 
which is the required rate of gain. 


Example 4. A broker bought stock at 3% discount ~ 


and sold at 8% premium, and gained $420; required 
the face of the stock bought. 
Form. 

Since he bought at 3% of the face less than the face 
and sold for 8% of the face more than the face, he 
gained 6% of the face, but $420 equaled his gain ; hence 
6% of the face of the stock equaled $420, and 1% of 
the face = % of $420 = $70, and 100% of the face = 100 
times $70 = $7000... ete. 


[Give other forms. ] 


Exercises. 
1. What cost 60 shares of railroad stock at 4% 
premium ? . 
2. Midland stock bought at 3% discount was sold 
at 4% premium; required the gain on 75 shares. | ~ 
3. 9 shares of I. & St. L. stock at 96 are exchanged 


for mining stock at 104; how many $100 shares of min- 
ing stock are purchased ? 


‘ ; ‘ 


PROBLEMS. 157 


4. The net earnings of a street railway are $1500; 
the capital invested is $30000 ; required the rate of div- 
idend that can be declared. 

5. How many shares of canal stock at 94 can bo 
bought for $118800 ? 

6. At 20% het how many shares of stock 
will #4800 buy ”? 

7. If Vandalia stock is quoted at 102} how much 
stock can be bought for $3246, brokerage being #% ? 

8. Pan Handle stock bought at 110 and sold at 116 
yields what rate of gain? 

9. Wm. Smith receives $630 as a 7% dividend; 
how many #50 shares does he own? 

10. A man owned 25 shares of rolling mill stock of 
$50 each; the company declared a dividend of 8%, 
payable in stock; how many additional shares were 
issued to him ? 

11. A bridge company whose stock was $12500 re- 
quired an assessment of $625; what rate did they de- 
clare ? 

12. Mr. Wells owns 35 shares of $100 each in a turn- 
pike company; his dividend was $201.25 ; required the 
rate of dividend. 

13. Jones bought 21 shares of mining stock at 15% 
discount; he sold it at an advance of 124% ; what did 
he gain, brokerage $%, each way? | 

14. I invested $3249 in stock of the Hartford Insur- 
ance Company, at 9} per cent. discount; how much 
stuck did I buy? 

15. A company which has a capital of $480000 has 


158 PERCENTAGE. 


earned $29540 net in six months;. it has declared a 
semi-annual dividend of 5%, and ae reserved the re-, 
mainder as a surplus; what is the surplus, and what is 

the dividend on 25 shares ? . 


16. A company with a capital of $250000, pays 35% - 
of its gross earnings for expenses, it has reserved $4365 
and has paid a dividend of 7%; what were its gross. 
earnings ? | 


17. I bought 100 es of ae na stock at 274; paid 
3 instalments of 15% ; I sold at 20% discount ; what 
did I lose on my investment ? 


18. A man bought 95 shares of rolling mill stock at 
1053, brokerage being $% ; he received a dividend of | 
54%; he sold at 1033; brokerage 4% ; what per cent. 
did he make on the investment ? 


- 19. When gold is quoted at 1024, how much paper 
currency can be bought for $200 in ed: no allowance’ 
being made for brokerage ? 


20. If the passage to Liver pool is $125 in gold, and 
gold is at 1034, what shall I pay in “ greenbacks” for | 
two tickets ? 

21. When gold is quoted at 103, what per cent. of a 
gold dollar is the value of a one dollar bill ? J 


22. If a man buys 20 shares of stock, originally 
worth $50 a share, at 10% discount, and sells at a 
premium of 8%; what does he make? 

23. When 75 shares of stock, originally worth $100 a: 
share, sells for $7556.25, at What per cent. above par 
does it sell ? 

24. A broker bought railroad stock at 80, and sold: 
at 112; what per cent. did he make? 


a 


INSURANCE. 159. 


219. Insurance. 
CORRESPONDING TERMS, 


1. The sum insured is base. 


2. The premium or the sum insured + the premi- 
um is a percentage of the sum insured. 

3. The rate of premium or +2° + the rate of pre- 
mium is the rate per cent. 

Remark. If the premium is viewed as the percentage, the 
rate of premium is rate per cent.; but if the amount or differ- 
ence be viewed as percentage the rate per cent. is }(¢ — 
the rate of premium. 


~ 220. Forms of Solution. 


Examples. 1. A house is insured for 800 at 4% ; re- 
quired the premium. 
2. The premium paid for insuring a barn for $1200 
is $6; required the rate of insurance. 
3. A man paid $150 for insuring goods at 3 yer 
cent.; required the value insured. 
Remark. The above problems come within Cases I, II and 


III, respectively, and are readily solved by forms similar to 
those given under the preceding applications of percentage. 


4, A stock of goods worth $19600 is insured at 2%, 


so as to recover both the value of the goods and the 
premium in case of loss; required the sum insured. 


First form. 

Since the premium is 2% of the sum insured, the 
value of the goods, $19600, must be 98% of the sum 
insured. We have given $19600, the value of the 
goods, which is percentage, and .98, the rate of differ- 
ence, which is rate per cent., to find the sum insured, 
which is base. Since the percentage is the product, «ec. 


[60 PERCENTAGE, 


Second form. 

Since 2% of the sum insured = the premium, 98% 
of the sum insured must = the value of the property 
=$19600; then 1% of the sum insured = #5 of $19600 
= $200, ian 100% of the sum insured = 100 times 
$200 = £20090... etc. 


5. A shipper took out a policy for $34200 to in- 
clude the value of the goods shipped and also the pre- 
mium at 6 per cent.; required the value of the goods. 


Form. 

Since the premium was 6% of the face of the policy, 
the value of the goods must have been 94% of its face. 
We have given the sum insured, $34200, which is base, 
and the rate of difference, .94, which is rate per cent., to 
find the value of the goods, which is percentage. 

Since the percentage is the product of the base and 
the rate per cent, etc. 

[Give other Foca 

Exercises. 
1. Property is insured for $2250 at 34% ; what is 
the premium? | 
2 <A stock of goods is insured for $4800 at 2%; 
what is the premium? 


3. A man pays $24 for insuring a house at 4 per 
cent; what is the face of the policy ? 

4. The rate of insuring a house was 14% and the 
premium was $12; what was the face of the policy ? 

5. $56 is the premium paid on a policy of $11200; 
what is the rate? 

6. A shipment of grain worth $15600 is insured at 
2%, so as to include the premium as well as the value 
of the grain in case of loss; what is the face of the 


policy ? 


TAXES, 161 


7. The rate of premium is 34% and the value of 
property insured is $2488 ; what face of policy will in- 


clude both in case of loss? 
_~> 8. To include the premium at 13% and the value 


of the goods, the face of a policy is $6300; what is the 
value of the goods? 

9. A vessel’s cargo is insured for $17600; the pre- 
mium at 4% is included; what is the value of the 
cargo ? 

10. What will it cost to insure a house for $900 at 
14 per cent? At # per cent? 

11. A piece of property was insured for $5000; the 
premium paid was $30; what was the rate? 

12. At 2}% what sum must be insured on property 
worth $3548 to include both property and premium in 
case of loss ? 

13. A merehant insured a consignment of goods for 
$15728 at 2% so as to include both property and pre- 
mium ; required the premium 

14. If a man pays $38.80 for insuring % of the value 
of his house, what is its value if the rate of insurance 
is 24%? 

15. A building is insured so as to include ? of its 
_yalue and theentire premium. The value of the build- 
ing is $24500, and the rate of insurance 14% ; what is 
the premium ? 


220. Taxes. 
CORRESPONDING TERMS. 


1. The assessed value of the property taxed is base. 
2. The rate of taxation is rate per cent. 
3. The tax is the percentage. 


Remark. Use forms of solution similar to those already 
given. 


162 PERCENTAGE. 


Exercises. 


1. What sum must be assessed in order that $12590 
may remain after paying a commission of 4% for cal- » 
lection ? | 

2. The valuation of the property in a certain dis- ; 
irctis #$z86474° A tax of $12560 is required. What 
tax must that man pay whose property is assessed at 
$8000 ? | : 

3. If the rate of tax was $12 on $1000 and the tax 
levied was $14674, what was the valuation ? 
4. Required A’s tax on property worth $2460 at 

5. What sum must be assessed at 3% to raise a tax 
of $7400, and pay a commission of 2% for collection? 

6. In a school district a school is maintained by a 
tax on the property of the district, which is valued at 
$648750. A teacher is paid $45 per month for 6 months, 
and other expenses are $64.50; required the tax on 
property worth #4865. 


7. Find A’s tax from the following items: 


His district paid in teachers’ salaries ........ $1200.00 
f nS fortiielenc ses Meee PEL: 57.60 
J 4 * incidentals) a. eee 38.00 


The money received from the school fund was $258 . 
The remaining expense was paid by arate bill. The 
aggregate attendance was 9568 days, while A sent 4 
pupils 46 days each. 


8 The cost of a public work was $1260. The rate 
. of taxation was 3 mills on the dollar, and the collector’s 
commissiom was 33% ; what was the valuation ? 


9 What is the value of property in a district assess-_ 


CUSTOMS... 193 


ing a fax. of $8540, if John Smith, who is wor th 
$12 180, pays a tax of $93.60? 

10. J. A. Williams’ income is $5480, of which $1860 
isexempt by law. After deducting house rental £840, | 
previous taxes $58 50, and other SrA CHOT $460, what 
is his income tax at 23% ? 


22. Customs: or Duties. 


CORRESPONDING TERMS. 


1. The net quantity of Cees is the base in Poe hile 
ing specific duties. 

2. The cost of the goods in the country whence they 
were exported is the base in computing Ad Valorem 
duties, a ea 

Remark. Duties are assessed upon the goods actually im- 
ported. All deductions are made previous to the assessment. 

3. The rate of duty is rate per cent. 

4. The duties assessed constitute the percentage: 

Remark. Use forms of solution similar to those already 
given. 
Exercises. 

J. Animportation was 56 casks of wine, each con- 
taining 36 gallons. The net duty at 30% Ad Val- 
orem, amounted to 3907, 20; required the invoice per 
Baliga. | 

2. A quantity of lace was invoiced at $816.54. The 
merchant paid the duties and a freight bill of $22.50, 
and found that the total cost was $980.50 ; required He 
rate of duty. 

3 Required. the duty at 24 cents per pound on 2700 
- pounds of cloves, tare being 5%. 


1O4 | PERCENTAGE, 


4. If the duty on opium is 100%, required the im- 
port tax on 236 lb. of opium invoiced at $3.75 per lb. 

do. Required the Ad Valorem duty at 30% on 125 
boxes of tea, each containing 70 lb. and invoiced at 
85¢ per lb., tare being 8.1b. per box 

6. A wine merchant paid $421.20 duties at 30%, 
leakage 10%, on 15 casks of wine invoiced at $2 per gal. 
How many gallons per cask were shipped? 

7. I paid $5670 duty at 25%, leakage 4%, on 50 
barrels of syrup, gross quantity 314 gallonseach. What 
was the invoice per gal ? , 


Applications of the Second Class. 


222. Interest. 
CORRESPONDING TERMS. 


1. The principal is base. 

2. The rate of interest is rate per cent. The rate of 
the amount or of the difference, 1. e. +99 + the rate of 
interest may be rate per cent. 

3. The interest is percentage. The amount is alsoa 
percentage of the principal. 

4. The number representing the time in years is a 
factor used with the principal and the rate of interest 
to determine the interest. 

Remarks. 1. The time unit in interest is 1 year of 12 months 
of 30 days each. 


2. Interest has involved in it four terms, viz.:—the principal, 
the rate of interest, the time in years, and the interest. "The 
first three of these are factors of the fourth. 


INTEREST, $65 
223. Case I. 


Given the principal, the rate of interest and the time 
to find the interest. 


Example. Required the interest of $500 for 2 yr. 
3 mo. 15 da, at 6%. 


First form. 
2 yr. 3 mo. 15 da.= 2,54 yr. 


Since the interest is the product of the principal, the 
rate of interest and the number representing the time 
in years, the interestin thisexample is found by multi- 
plying together $500, O6and2,4. The product, $68.75, 
is the required interest. 


Remark. The following expressions for the above form indi- 
cate a method for finding interest for months or days: 


es onthe. ! oes Sen in months _1, teen 
Bor fave EEE ta x time in CayS. or toreah 
1230 
Second form. 

hy of #500 for 1 yr. at 6%:= $30.00 
* p00) “> Tmo.“ 6%=7, of . $80— 2.50 
eno) <1 da“ 69-2, of $2.50— wie 
Pree OU ad yt. 6 ==2 times $30 == $60. 00 
Pe wan) 'o m0. 6%e=p- *.-$2.50,.< 750 


“« «& 500 “15 da. “6%=15 “ OBR a Sb 
Peon 2 yr, +3 mo, 15 da at6Ge= $68.75 


166 PERCENTAGE. 


Exercises. 

1. The principal is $150, the rate of interest 6%, 
th: time 3 yr.; required the interest. 

2. Find the interest of $750 for 2 yr.8 mo. ats %. 

3. Find the interest of set for 5 Mh 10 mo. 15 da 
at 8%. 

4. Find the interest of $75 for 11 mo. at 9%. 

5. Find the interest of $956 for 7 mo. at 10%. 

6. Find the interest of $1278 for 5 mo. 3 da. at 6 per 
cent. 

7. Find the interest of $16 for 90 da. at 12%. 


8. Find the interest of $800 for 5 yr. 4 mo. 20 da. at 
10 per cent. 


9 Find the interest of bie to Sor 6 yr. 2 mo. 28 da. 
at 8 per cent. 


10. Find the interest of $9.50 for 20 da at 10%. 
11. Find the interest of $17850 for 40 da. at 11%. 


12. Find the interest of $675 from July 3, 1881, to 


August 7, 1883, at 8 per cent. 

13. Find the interest of $95 from May 10, 1883, to 
April 6, 1886, at 7 per cent. 

/14. A note was drawn for $850 on January 8, 1885; 
a payment of $200 was made September 18, 1885; 
what was due January 8, 1886, rate of interest being 
5 per cent.? 

15. A note was drawn for $1000 on June 6, 1878; 
its rate of interest was 6%. A payment of $450 was 
‘made June 6, 1879; another of $200, September 21, 
1879; another of $350, May 10, 1880; what was due De- 
. Cem lee 25, 1881? 

16. A note was drawn for $67 on Obtobet LA Tes2 


INTEREST, ‘107 


the rate of interest being 9%. A payment of $16 was 
‘made Decen. ber 1, 1882; another of $22, March 12, 
1883 ; another of $18, June 11, 1883; another of $7, 
September 17, 1883; what was due October 17, 1883? 
17, What is the compound interest of $156 for 4 yr. 
6 mo. at 6 per cent.? 
18. What is the compound interest of $1350 for 3 yr. 
8 mo. at 6 per cent.? 
19. What is the compound interest of $1100 for 1 yr. 
6 mo. at 8% ; interest compounded quarterly ? 
20. What is the compound interest of $800 for 3 yr. 
at 5% ; interest compounded semi-annually? 
21. What is the annual interest of $750 for 3 yr. 4 
mo. at 6 per cent ? 
—— 22. Find the annual interest accruing on $65 for 5 
yr. 8 mo. 12 da. at 7 per cent. 


224. Case Il. 

Given, the principal, the time and the interest, to 
find the rate of interest. 

Example The principal is $500, the time 25% yr. 
and the interest $68.75 ; required the rate. 
First form. 

Since the interest is the product of the principal, 
the rate and the number representing the time in 
years, the rate of interest in this example is found by 
dividing $68.75 by the product of $500 and 2,4, and 

making the qnotient hundredths. The quotient, .0C, 
is the required rate. 
| Second a 


_ The interest of $500 for 2,4 yr. at 1%=$1144. Now 
since $1144 = the interest 6 $500 for res yr. a 1% 


idl a ge hig > a YR 
oe ee tr NAD Eee 
Cf, aT me come ette Sk 


168 PERCENTAGE. 


$68.75 =the interest of $500 for 2,4 yr. at as many 
times 1 % as $68.75 is times $1141 which=6. 6 times 
1 % =6%.. .*. the required rate of interest is .06. 
Exercises. 

1. The principal is $380, the time 3 yr. 4 mo. and 
the interest $76.76; required the rate. 

2. A man received $218.40 interest on a loan of 
$780 for 4 yr. 8 mo.; required the rate. 

3. The principa! is $4760, the time 2 yr. 8 mo. 20 
da. the interest, $864; required the rate. 

4. The interest is $456.84, the time 5 yr. 9 mo. 15 

da.; what is the rate, the principal being $986? | 

5. The principal was $960, the time 7 yr. 5 mo. and 
the interest $520.80; required the rate. 

6. The principal is $480, the time 6 yr. 3 mo. and 
the interest $210; what is the rate ? 

7. The time is 8 yr. 9 mo. 12 da., the interest 
$17.46, and the principal $26.50 ; required the rate. 

8. The principal is $2015, the interest €575.40 and 
the time 5 yr. 8 mo. 15 da.; required the rate. 
225. Case Ill. 


Given the principal, the interest and the rate, to 
find the time. 

Example. In what time will $500 yield $68.75 at 
6%? 

First form. 

Since the interest is the product of the principal, 
the rate and the number representing the time in 
years, the time in this example is found by dividing 
$68.75 by the product of $500 and .06. The quotient, 
254, 1s the number representing the time in years. 
.. the time is 254 yr., or 2 yr. 3 mo. 15 da. 


INTEREST, 169 


Second form. 

Since $500 at interest for 1 yr. at 6% yields $30, 
$500 must be on interest for as many times 1 yr. at 6% 
to yield $68.75 as $68.75 are times $30, which=2,;%. 
20% times 1 yr.=254 yr.=2 yr. 3 mo. lo da. .°. ete. 


Exercises. 


1. The principal is $4080, the interest $668.10, and 
the rate 5% ; required the time. 

2. The principal is $176, the interest $22 ; and the 
rate 7% ; required the time. 

3. The principal is $1300, the interest $274 and the 
rate 8% ; required the time. | 

4. How long will it take any principal to double 
itself at 4% ? at 5% ? at 10% ? 


226. Case IV. 


Given the interest or amount, the time, and the rate 
of interest, to find the principal. 

Example. What principal will yield $68.75 int. in 
2 yr. 3 mo. 15 da. at 6% ? 


First form. 

Since the interest is the product of the principal, the 
rate of interest, and the number representing the time 
in years, the principal in this example is found by 
dividing $68.75 by the product of .06 and 254. The 
quotient, $500, is the required principal. 


— Second form. 

A principal of $1 will yield $.13? interest in 2,4 yr. 
at 6%; and to yield $68.75 interest in the same time 
at the same rate would require a principal as many 
times $1 as $68.75 are times $.133, which = 500. 500 
times $1 = $500. .°. etc. 

‘12 


170 PERCENTAGE. 


Exercises. 
1. Required the principal if the time is 8 yr. 8 mo., 
the rate :06, and the interest $462. 
2. Required the principal if the time is 6 yr. 3 mo., 
the rate .07, and the interest $64.26. 
3. What principal willin 7 yr. 4 mo. at 8% amount 
to $749.70? 
4. What sum will yield $185 int. in 18 mo. at 5%? 
5. What.sum must be invested in 6% stocks to 
yield an income of $1500? 
6. What principal will amount to $200 in 14 yr. 3% 
mo. at 7% ? 
7. What principal will amount to $355. 60 i in 2 yr. 


7 mo. at-8% ? 
7 Review Exercises in Interest. 
Principal. Rate. Time. Interest... Amount. 

1. $420 6% 3 yr. 6 mo. ? ? 
2. 492.15 6“ 1yr.38mo.18da ? ? 
3. ? 6“ 4 yr. ; $24 ae 
4. B00 Oe KO? 36 ? 
5. ? 6 “ 1 yr. 6mo. Wee fe Qin 
6. ? 8 * 2 -yr..11 mo. 27 da. $845 
{¢ 12.80 7 3yr. 4 mo. 3 da. ? 2 
S5 eae. 5“ 2yr.4mo.12da. 40 
9. 750 Sm ie 120 

10. 3542 (i2 yr. Cuno. 442.75 

11. ? ot Or Sao. 900 

12. 1475 10“ 5 yr.3 mo. 5 da. ? 

13. 50 Sor Re 20 | 

14, 500 ? “ 2 yr. 6 mo, 50 

aya 63 yri2m0; 5 
16. ? 66°02 vriaG m00: 76 
17: 1000 4“ 1 yr.8 mo. . ? 


18 02 5 “ 30 da. eae 2072 


"DISCOUNT. 171 


Principal. Rate. Time. — Interest. Amount. 
Peo.) 0“ Sryr. 2 mo, 2V da. |‘? ? 
20. 176.25 ‘?** lyr. 11mo.5da. 25.52 
21. 185.85 34“ 3 yr. 5 mo. 15 da. 2 
eaten. t 12 ** 90 da. 412 
23. Ge Si yr. Omo. ? 690 
24. : q “-6ryr, 157.50 ? 
25. Ths 8 “ lyr. 6mo. 24 da. 30.24 


26. 82.50 6“ dS yr. 8 mo. 12 da. ? 
27. 450 2?“ 1Lyr.8mo.12da. 61.20 


98. 600 9* ? 798 
> EEG Y rae Oma 90 
See 5B yr} 341.25 
eh Cn 68.60 


reo. Ue 924.70 


227. Discount. 


Discount is treated under three heads, viz: True 
Discount, Bank Discount and Commercial Discount. 


a. True Discount. True Discount is an application 
of Case IV in interest. 


CORRESPONDING TERMS. 
1. In true discount the present worth is the principal. 


2. The debt is amount, and may be considered as 
percentage if the time element be reduced to 1 yr. 


Remark. The rate of interest and the number representing 
the time in years being factors, if one of them be divided and 
the other multiplied by the same number, the value of the pro- 
duct is not affected. e.g. Ifthe time should be 3 yr. and the 
rate .06, we may reduce the time element to 1 yr. by dividing 
it by 3 if at the same time we multiply the rate, .06, by three 

It is evident that the interest of a given principal for 3 yr at 
6 per cent. equals the interest of the same a Se for 1 yr at 
18 per cent. 


172 PERCENTAGE. 


3. The discount is the percentage. (Interest). 

4. The rate of interest is rate per cent. The relation 
(expressed in hundredths) of the debt to the present 
worth may be the rate per cent. 


5. If the debt be considered as percentage, the rate 
per cent. becomes +92 + the given rate multiplied by 
the number oramnuine the time in years; the time 
element being divided by itself and thus reduced to 1 
year. 

6. The time named is the time element, except as 
stated in 2 and 5. 

Example. Required the present worth and true dis- 
count of $568.75 due in 2 yr. 3 mo. 15:da. if money is 


worth 6%. 
First form. 


The time element is reduced to 1 yr. by dividing it 
by 254. By multiplying the rate, .06, by 2,4, it becomes 
133. 

If the debt, $568.75, be considered as percentage, the 
rate per cent. becomes 1.132. 

Wethus have given the percentage, $568.75, and the 
rate per cent., 1.132., to find the base, which is the 
required present worth. 

Since the percentage is the product of the base and 
the rate per cent., the base in this example is found by 
dividing $568.75 bY 1.13%. The quotient, ati is the 
required base, or present worth. 


Second form. 
$1 at interest for 2 yr. 3 mo. 15 da. at6%, amoun’s 
to $1.13%. $1 is, therefore, the present worth of $1.13? 
due in 2 yr. 3 mo. 15 da. without inter est, money being 
worth 6%. Hence the present worth of $568. 791 as 
many times $1 as ane ie are times $1. 133, whicl. = 
500. 500 times $1 = $50 ba ro 


DISCOUNT, 173 


Exeroaises, 


1. Required the present worth and true discount of 
¢136 due 3 years hence, if money is worth 12%. 
2. A note of $4800 is due in 4 years. What is its 
cash value if money is worth 5% per annum? 
4&8. A was offered a lot for $225 cash or $230 in 3mo. 
Did he make or lose, and how much by accepting the 
latter offer? 2704024 ee 


4. Required the discount of a pap of 9864 due in 
8 mo. if paid now? , 

5. A grocer bought 62 baila of eoleens of 314 
gal. each, at 26¢ per gal., on 90 days’ time, and sold 
emiedintely for $615; how much did he gain if money 
was worth 8% ? 

6. What is the present worth and true discount of 
$27.50 due in 20 months, if money is worth 8% ? 

7. Hogs were purchased to the value of $1574, one- 
half payable in 3 mo. and the remainder in 6 mo., 
without interest. What is the cash value of the stock 
if money is worth 7%? 

8. Which is worth the most $640 in 12 mo., $620 
in 6 mo. or $600 in cash, if money is worth 9% ? 


b. Bank Discount. Bank Discount is the simple 
interest of a given sum for the time elapsing between 
the date of discounting and the date of legal maturity. 
This time is called the term of discount. 


CORRESPONDING TERMS. 
1. The face of the obligation is principal or base. 


2. The rate of discount, or rate of proceeds is rate 
per cent. 

8, The discount or proceeds is percentage. 

4. The term of discount is the teme element. 


174 PERCENTAGE, 


Remnrks. 1. If any interest bearing note be discounted ata 
bank, the amount of the note is found at the given rate of inter- 
est for the time elapsing between the date of the note and its 
legal maturity. The amount thus found is then discounted at 
the rate of discount for the term of discount. 


2. If the face of the note, (principal), the rate of discount’ 
(rate of interest), and the term of discount (the time) be given 
to find the discount, (interest), the problem is solved under 
Case I in interest. 


3. If the face of a note, (principal), rate of discount, (rate of 
interest), and term of discount, (time element), be given to 
find the proceeds, the discount, which is interest, is found un- 
der Case I in interest. The discount is then subtracted from 
the face of the note; the remainder being the required pro- 
ceeds; or the rate of proceeds may be used as rate per cent> 
and the problem solved directly by Case I in interest. 


4. If the proceeds, rate of discount or rate of proceeds, and 
term of discount be given to find the face of the note, the 
problem is solved under Case IV in interest, considering the 
proceeds as interest, and the rate of proceeds as rate of interest, 
first reducing the time element to 1, and increasing the given 
rate of discount in the same ratio. 


Example. Given the proceeds $498, the term of 


discount, 63 days, and the rate of discount, .08, to find - 


the face of the note. up 


Solution. The time element 3°3,, divided by itself 
is reduced to 1. 

The given rate of discount, .08, multiplied by 83, 
=,.012. 

The rate of proceeds is, therefore, .983. 

The following form exhibits the work. 


493 
aay 8500= the face of the note. 


[Nore.— For practical purposes the form of solution usually 
given in the text books for the above problem is preferable to 
the solution here given. ] 


DISCOUNT. {75 


Exercises. 


1. Find the bank discount and proceeds of a note 
of $50 payable in 60 days, discounted at 8%. 

2. A note of $56 dated Jan. 1, 1884, and payable 
May 1, 1884, is discounted in bank at 6% ; required the 
proceeds. , 

3. A note of $500, dated Dec. 15, 1883, and payable 
Feb. 18, 1884, is discounted at 9% ; required the pro- 
ceeds and the discount. 

4, A note of $650 with interest at 8% is dated Nov. 
80, 1883, and is payable in 90 days. It is discounted 
Jan. 5, 1884, at 10% ; required the proceeds. 

5. A note of $1400 with interest at 10% is dated 
Jan. 16, 1884, and is payable May 18, 1884. It is dis- 
counted April 7, 1884, at 12% ; required the proceeds. 

6. For what sum must a 60 days note be drawn 
that when discounted in bank at 6% the proceeds may 
be $1000? 

7. A owes B $1500; for what sum must a 90 days 
note be drawn that when discounted in bank at 6%, 
B may obtain his money ? 

8. A merchant bought goods for $1621.20 cash, ob- 
taining the money from a bank on a 60 days 7% note; 
what was the face of the note? 

: 9. Required the cash value of a note of $6780 dis- 

counted in bank for 4 mo. 15 da. at 6%. 

“10. Required the face of a note given in bank that 
when discounted for 5 mo. 21 da. at 7%, the proceeds 
shall be $57.97. 


c. Commercial Discount. Commercial Discount is a 
decuction from the face of a bill or other obligation 
without regard to time. Itis also called per cent. off, 
and is effected under Case I in Percentage. 


176 "PERCENTAGE. 


Exercises. 
1. A merchant bought $1200 worth of goods on 6 
mo. time; but paid cash on obtaining a discount of 8% 
off. What did he pay? 


2. 6% off was allowed on a bill of goods amount- 

ing to $1878.50; what sum did they cost ? 

+8. A country merchant purchased a bill of goods 
amounting to $3675 on 4 mo., but was offered 5% off 
for cash. Would he gain by borrowing the money 
from a bank at 8% per annum for the time and pay- 
ing the cash? 

4, What is the cash value of goods listed at $5650, 
10% off for wholesale and 5% off for cash ? 

5. A merchant paid $1.14 per yd. for goods after a 
discount of 6% had been made from the marked price. 
What was the marked price? 

6. What was the invoice price of goods for which I 
paid $39 after a discount of 40% had been made? 


228. Exchange. 
CORRESPONDING TERMS, 
t. The face of a draft is base. 
2. The exchange or the cost of a draft is percentage. 


3. The rate of exchange or the rate of cost is. rate 
per cent. The rate of cost is also called the course of 
exciiange. 

4, In time drafts the time named plus 3 days is the 
tizne element. 

Renarks. 1, Theinterest accruing on a time draft isdeducted 
frcm the face of the draft, for, the bank having the use of the 
money during the time should, in equity, pay the interest. 

2. Make each problem an application of either the first or the 


s-cond class of the applications of percentage, according as the 
element of time is or is not involved. 


EQUATION OF PAYMENTS. 177 


Exercises | 

1. Required the cost of a draft for $378 on New 
York at }% premium. | 

2. What is the cost of a draft for $780 on Chicago 
at 2% discount? 7 

3. Required the cost of a draft for $560 payable 30 
days after sight, exchange 4% premium, and interest 
6%. 

4. What is the face of a 30 days draft which 
cost $352.62, exchange being 14% discount, and int- 
erest 6 per cent.? 

*5. What is the face of a draft on Indianapolis at 45 
days, costing $145 ; exchange at 3% premium? 

6. How much must I pay in Paris for a draft on 
Chicago for $4500 at 182¢ per franc, at 1% premium ? 

7. What must be paid in New York for a draft on 
London of 560£ at 8% premium ? 

8. What is the face of a 60 days draft that cost 
$1000, exchange $% discount, and interest 6% ? 

9. Required the cost of a 30 days draft for $1920 at 
#7, discount, interest 7%. 

10. What is the face of a 60 days draft that can be 
bought for $3195.20, interest 8% and exchange 14% 
premium ? 


229. Equation to Payments. 


Remark. A problem in equation of payments usually con- 
tains a number of problems under Case I in interest; the ob- 
ject being to find an equitable time for the payment of several 
sums of money due at different times. 


178 PERCENTAGE. 


Exercises. 
“A 


Remark. For methods of soluticn see text books. 

1. Required the average term of credit for the fol- 
lowing debts: $400 due in 3 mo., #500 due in 5 mo. and 
$700 due in 8 mo. 

2. A debt of $2400 is subject to the following con- 
ditions: $800 is due in 4 mo., $600 is due in 6 mo., and 
the remainder is due in 8 mo. What is the average 
term of cr-dit ? ; 

~8. A man owes $240 due in 20 days, and $560 due’ 
in 30 days. At the end of 16 days he pays$300 and at 
the end of 24 days he pays $3850; when, in equity, 
should he pay the remainder ? 

4. A merchant bought goods as follows: April 1, - 
$280 on 3 mo, time, $200 on 4 mo., $300 on 5 mo., and 
$560 on 6 mo. On what date will a single payment 
discharge the debts? 

5. Wm. Smith owes $30 due in 60 days, $100 due 
in 120 days, and $750 due in 180 days; what is the 
equated time of payment? 

_~6. Mr. Wallace bought grain on a credit of 90 days 
to the following amounts: 


25th: of Jan wsioe ek Le eee $3750 
10th of Peabo ue yey ox eee 3000 
6th Of - Mareho oe eres a ee 2.400 


On the first day of May he wishes to give his note 
for the amount. At what time will it mature? 


7. A merchant bought goods as follows: Feb. 10, 
- $364; March 12, $375; April 15, $554; May 18, $622. 
He obtains 6 months’ credit on each purchase; at what 
time can the whole be equitably discharged ? Sn 

8. What is the average date for paying three bills, 
due as follows: Jan. 31, $477; Feb. 28, $377; March 31, 
$777? 


EQUATION OF PAYMENTS, 179 


9. Required the average time of the following bills, 
allowing to each term of credit 3 days of grace. April 
3, $500 on 3 mo.; April 4, $200 on 2 mo.; April 4, $200 
cash ; and April 10, $500 on 3 months. 

10. Find the equated time for the payment of the 
following notes: $350 dated July 12, 1883, for 60 days; 
$720 dated Sept. 10, 1883, for 90 days ; and $1200 dated 
Nov. 5, for 120 days. 


11. A debt is to be paid as follows: One-sixth now, 
and one-sixth every 3 months until all is paid. When 
might the entire debt be paid at once? Ans. 74 mo. 


12. C owes D $800, one-sixth of which is to be paid 
in 2 months, one-third in three months, and the re- 
mainder-in six months; but he proposes to pay one- 
half now. When in equity should he pay the remain- 
ing half? Ans. 94 mo. 


13. A merchant bought goods amounting to $10500 
payable in 6mo. He paid $2000 in 2 mo. and $4500 
in 4 mo; how much time is he entitled to on the re- 
mainder? Ans. 104 mo. 


14. When will a single payment discharge the fol- 
lowing: 
Bought goods—June 1, $500 on 3 mo. credit. 
PUY Ds SOOO Shhh oS 
AMIE GB, SOULE e ohetey ik nth 
Oat. Se S800 Oth i ae 
Ans. 52 da. after Sept. 1. 


15. A owes $600 due in 6 months, but at the end of 
8 months he desires to make a payment sufficiently 
large that the remainder may not be payable until 6 
months after its first date of maturity ; how large must 
be the payment ? 


CHAPTER XIIL 


RATIO AND PROPORTION. 
Ratio. 


230. Ratioisthe relation of one number to another 
considered as a measure. 


Example. The ratio of 6to3 = 2;1.e. if 3 be ap- 
plied as a measure to 6, the number of applications that 
can be made is 2. 2 is, therefore, the relation that 6 
sustains to 3 considered as a measure. 


The ratio of 1 to 2 = 4; of 2 to 3 = 4, etc. 


Remarks. 1. The number of times that a divisor is contained 
in a dividend is the ratio of the dividend to the divisor. A 
quotient if abstract, may be viewed as a ratio. 

A multiplier is the ratio of the product to the multiplicand. 


2. The part that one number is of another is called the ratio 
of the first to the second. Ratio is thus related to fractions in 
that it is the relation of a part to a whole. 


3. Since ratio is the relation of measure, the two numbers byes 
tween which a ratio exists are like numbers 
23!. The Terms Used. Three terms are concerned 
in thinking a ratio: viz.—a dividend, a divisor, and a 
quotient. These are called, respectively, Antecedent 
Consequent and the Ratio. 

Antecedent. The dividend, or first term of a ratio is 
called the antecedent. 


Consequent. The divisor, or second term of a ratio is 
called the consequent. 


The Ratio. The quotient of the antecedent by the 
consequent is called the ratio. 


RATIO. 181 


232. The Notation of a Ratio. 


A colon is used to separate the written terms of a 
ratio, thus— 6:3. This expression is read the ratio of 
6 to 3; it expresses the quotient of 6 by 3. 


233. Classes. 


Simple. A ratio each of whose terms is a single 
number either integral, fractional or mixed is called a 
single ratio. As— 6:2; 4:4; 24: 34. 


Compound. Two or more simple ratios, viewed to- 
gether as to the product of their corresponding terms, 
constitute a compound ratio. 


234. Principles. 


Remark. Since the terms antecedent, consequent and ratio 
signify dividend, divisor and quotient, respectively, the general 
principles of division become, by a change in terminology, the 
principles of ratio. 


I. Multiplying the antecedent multiplies the ratio. 


_ IL. Multiplying the consequent divides the ratio. 


III. Multiplying both antecedent and consequent 
by the same number does not change the ratio. 


IV. Dividing the antecedent divides the ratio. 
V. Dividing the consequent multiplies the ratio. 


VI. Dividing both antecedent and consequent by the 
same number does not change the ratio. 


- Remark. Prin. I1Il may be thus stated: The ratio between 
like multiples of two numbers equals the ratio between the two 
nuinbers. Prin. VI may be thus stated: The ratio between 
like parts of two numbers equals the ratio between the two 
numbers. | 


182 RATIO AND PROPORTION 


Proportion. 
234. Two equal ratios constitute a proportion. 


Remark. The ratios which form a proportion may both be 
simple, both compound, or one simple and the other com- 
pound. 

A Simple Proportion. A proportion that consists of 
two simple ratios is called a simple proportion. 

A Compound Proportion. A proportign containing a 
compound ratio is called a compound proportion. 
235. Notation. 


A proportion is notated re BB a double colon 


between the two equal ratios,and interpreting it by the 


word as. The proportion, 2:4::5:10,is read 2is to 4as 
5 isto 10. <A proportion is often notated by writing 
the two equal ratios with the sign of equality between 


them. The proportion, 2: 3=4: 6, is read—The ratio” 


of 2 to 3 equals the ratio of 4 to 6. 


Remark. ‘The antecedent of the first ratio and the conse- 
quent of the second ratio of a proportion are called the extreme 
terms and the other terms the mean terms of the proportion. 


237. Principle. 


The product of the mean terms of a proportion 
equals the product of its extreme terms. 


Demonstration. In a ratio the antecedent is the pro- 
duct of two factors, viz.: the consequent and the ra- 
tio. Ina propeviien the product of the mean terms is 
composed of three factors, viz.: The ratio and the 
consequent of the second ratio (which compose the an- 
tecedent of the second ratio,) and the consequent of the 
first ratio. 


The product of the extreme terms is composed of 
three factors, viz.: the ratio and the consequent of the 


Haw rae 


PROPORTION. 183 


first ratio (which compose the antecedent of the first 
ratio) and the consequent of the second ratio. 

It is thus observed that the factors composing the 
product of the means are identical with the factors 
composing the product of the extremes ; hence the two 
products are equal. 


238. Forms of Solution. 


Remark. 1. Any problem that is solvable by proportion is 

readily solved by analysis. 

2. It is to be remembered that a ratio exists between like 
numbers only. 

3. In solying a problem by proportion two distinct steps are 
_taken. 

a. The arrangement of the ratios, called the statement of the 
‘proportion. | 


b. The reduction of the proportion, or the finding of the un- 
known term. 


Example. If 11 bu. of wheat cost $9, what cost 
Py bu? 


Blackboard form. Thought form. 

9:x::11:17| Since a ratio exists between like 
numbers only, a ratio exists, in this example, between 
11 bu. and 17 bu., and between $9, the cost of 11 bu. 
and the cost of 17 bu., and these ratios must be equal. 
9, which rspresents the cost of 11 bu., may be made 
either term of either ratio, and the required number, 
which we will represented by x, will be the other term 
of the same ratio. We choose to make 9 the antece- 
dent of the first ratio; then is z the consequent of the 
first ratio. $9 is the cost of 11 bu., while $z is the cost 
of 17 bu., a greater number than 11, hence the conse- 
quent, x, of the first ratio is greater than its antecedent, 


Ce a Sy 
Vesa ghey as 


184 RATIO AND PROPORTION. 


9; and since the two ratios must be equal to forma 
proportion, the consequent of the second ratio must 
be greater than its antecedent. Hence we make 11 
(bu.) the antecedent and 17 (bu.) the consequent of 
the second ratio. 

We now have the two extremes and one mean of a 
proportion. Since the product of the extremes equals 
the product of the means, the required mean is found 
by dividing the product of the extremes by the given 
mean (using the numbers abstractly in performing the 
operation.) The required mean is 13.91. .. 17 bu. 
cost $13.91. 


Example 2. If 75 men can build a wall 50 ft. long, 
8 ft. high and 3 ft. thick, in ten days, how long will it 
take 100 men to build a wall 150 ft. long, 10 ft. high 
and 4 ft. thick ? 


Blackboard form. 


%:10::75:100 
150 : 50 
10:8 
4:3 
10 75150«10*4 
100«50«K 8x3 


Thought form. 


In this problem a ratio exists between 75 men and 
100 men; 50 ft. length and 150 ft. length; 8 ft. hight 
and 10 ft. hight; 3 ft. thickness and 4 ft. thickness; 10 
days and the required number of days which we may 
represent by 2. 


10 (days) may be made either term of either ratio, 
and x (days) will be the other term of the same ratio. 


375 


PROPORTION. ~ 185 


We will make 10 the consequent of the first ratio, then 
will x be the antecedent of the first ratio. 

10 days are required for 75 men to do a work while 
x days are required for 100 men to do the work. 100 
men require a less number of days than 75 men, hence 
the antecedent, x, of the first ratio is less than its con- 
sequent, 10; and since the two ratios must be equal to , 
form a porportion, the antecedent of the second ratio 
is less than its consequent; we therefore, write 75 as 
the antecedent and 100 as the consequent of the sec- 
ond ratio. . 

Again,—10 days are required to build a wall 50 feet 
long, while x days are required to build a wall 150 feet 
long, a greater length than 50 feet, hence z is greater 
than ten, and as the two ratios must be equal to form 
a proportion, the antecedent of the second ratio must 
be greater than its consequent ; we therefore write 150 
as the antecedent and 50 as the consequent of the sec- 
ond ratio. | 

Again,—10 days are required to build a wall 8 feet 
high, while x days are required to build a similar wall 
10 ft. high, a greater hight than 8 ft., hence x is more 
than ten, and since the two ratios must be equal to 
form a proportion, the antecedent of the second ratio is 
greater than its consequent; we therefore write 10 as 
the antecedent and 8 as the consequent of the sec. 
ond ratio. 


Again,—-10 days are required to build a wall 3 feet 
thick, while x days are required to build a similar wall 
4 ft. thick, a greater thickness than 3 ft., hence a is 
more than 10, and, since the two ratios must be equal 
to form a propertion, the antecedent of the second 
ratio is greater than its consequent ; we therefore, write 
4 as the antecedent and 3 as the consequent of the sec- 
ond ratio. 


ee) 


PD ety oat eee RP Rah 
re Fa etna ey | 4h 
oy ‘ 


186 RATIO AND PROPORTION 


We now have given the factors of the means and the 


factors of one extreme of a compound proportion to 
find the other extreme. | 


Since the product of the extremes equals the pro- 
duct of the means, the required extreme is found by 
dividing the product of the factors of the means by the 
product of the factors of the given extreme. The re- 
quired extreme is found to be 374. -... the required 
time is 374 days. 

Remark. The work may be shortened by considering the 


factors of the means as factors of a dividend and those of the 
giyen extreme as factors of a divisor, and canceling. 


Exercises. 


If 70 horses cost $3500, what cost 160 horses ? 

If 5. lb. of coffee cost $1.85. what cost 9 lb.? 

If 12 tons of hay cost $87, what cost 17 tons? 

. If a farm cost $4800, what cost 2 of it? 

. If § yd. of silk cost $1.65, what cost 8 yd.? 

. If 19 acres of land sell for $570, required the price 
of 40 acres. 


7. How many men will do as much work in 84 days 
as 8 men do in 126 days 2, 
8. If I borrow $3500 for 30 days, for what time may 
I return $900 to requite the favor 
9. If a 10 cent loaf weighs 1 Ib. 2 oz. when flour is 
$74 per barrel, what should it weigh when flour is $6 
per barrel ? 
10. What cost 147 bi. of corn if 35 bu., cost $284? 
11. If 15 sheep cost $15.90, how many sheep can be 
bought for $155.82 ? ) 
12. At 36d. per $4, what is 1£ worth in U. 8. 
money ? 


13. In what time can a man pump 54 barrels of 
water, if he pump 24 barrels in 1 hr. 14 min.? 


> OTR CoN 


— 


PROPORTION. ; 187 


14, If 10 bales of cotton can be carried 115 miles for 
$8, pon far can 15 bales be carried for the same 
mon3y” 

15. A mill makes 1265 barrels of flour by running 
10 hr. per day for 13 days; how many barrels would 
it make in 26 days, running 15 hr. per day ? 

16, If it cost $28 to carpet a room 12 ft. by 7 ft., 
what will it cost to carpet a room 30 ft. by 18 ft.? 

17. If 60 men dig a canal 80 rd. long, 12 ft. wide 
and 6 ft. deep in 18 days, working 16 hr. per day, how 
many men will be required to dig a ditch 30 rd. long 
9 ft. wide and 4 ft. deep in 24 days, working 12.hours 
per day? 

18. A man gained $3155 on the sale of 91 horses; 
how much would he have gained on 245 horses sold at 
the same rate? 

19. If a tank 174 ft. long, 114 ft. wide and 18 ft. 
deep, contain 546 barrels, how many brrrels will a tank 
hold that is 16 ft. long, 15 ft. wide and 7 ft. deep? 

20. If 8366 men in 5 days of 10 hr. each, can dig a 
trench 70 yards long, 3 yards wide, and 2 yards deep ; 
what length of trench 5 yards wide and 8 yards deep, 
can 240 men dig in 9 days of 12 hr. each? 

21. If it cost $64 to pave a walk 3 ft. wide and 382 ft- 
long, what will it cost to pave a walk 5 ft. wide and 64 
ft. long? 

22. If 7 men, working 10 hr. per da., make 6 wag. 
ons in 21 days, how many wagons can 12 men make 
in 16 days working 9 hours per day ? 

23. If it cost $320 to buy the provisions consumed 
in 8 mo. by a family of 7 persons, how much at the 
same rate will it cost to feed a family of twice the num- 
ber of persons for + as much time? 

24, Ifa stone 2 ft. long 10 in. wide, and 8 in. thick, 


188 RATIO AND PROPORTION 


weigh 72 lb., required the weight of a similar stone 6 
ft. long, 15 in. wide and 6 in. thick. 

25. If 8300 ib. of wool at 28¢ per lb. are exchanged 
for 36 yd. of cloth 14 yd. wide, how many lb. of wool 
at 35¢ per lb., should be given for 20 yd. of cloth # yd. 
wide ? 


Proportional Parts. 


239. A number is divided into proportional parts if 
separated into parts whose ratio equals the ratio of 
given numbers, called proportionals. 


Example. Divide 40 into two parts that are to 
each other as 3 to 5. 


Solution by Analysis. 


40 is to be divided into parts that are respectively 
equi-multiples of 3 and 5; i. e., one of the required 
parts of 40 is as many times 3 as the other part is 
times 5. 

40, the sum of the required parts is, therefore, the 
same number of timesthesum of 8and 5. 4(0is5 times 
the sum of 3and 5. 5 times 8=15 and 5 times 5=25. 
Hence 15 and 25 are the required parts into which 40 
is to be divided. 


Solution by Proportion. 


a. The sum of the proportionals, 8, is to the sum of | 


the required parts, 40, as the greater proportional 5, is 
to the Bpenler of the required parts. Thus: 8:40::5:2. 
-b. 8:40::3:2, the lesser part. 


poms a. The sum of the proportionals is to He 


Suan % the required numbers as the greater proportional is to 


‘eesunetf the greater of the required numbers. 


oe) 4 en 


PROPORTION. 189 


b. The sum of the proportionals is to the sum of 
the required numbers as the less proportional is to the 
less of the required numbers. 


Exercises. 


1. Divide 60 into two parts proportional to 11 
and 9. 

2. Divide 60 into three parts which are to one 
another as 2, 3 and 5. 

3. Itis required to divide 78 into three parts which 
shall be to each other as 3, 4 and 6. 

4, Divide 1275 into three parts which are to each 
other as 8, 5 and 7. 

5. The sum of two numbers is 60; the first is to the 
second as 5 is to 7; what are the numbers? 

6. D and E hired a horse for $40; D used it 6 wk., 
and H 4 wk.; how much should each pay? 

7. A man having 200 head of cattle, wished to 
divide them into three herds which should be to each 
other as 2,3 and 5; how many will each herd con- 
tain ? 

8. The sum of three numbers is 120; they are to 
one another as 4, 4 and =; what are the nner? 

_ 9. Two kinds of feed are mixed in the ratio of 15 
Ib. of meal to 9 lb. of bran. How mueh of each in 120 
lb. of the mixture? 

10. Divide $8470 into parts proportional to 3, 4,4 
and + 

Ste A, B and C agree to pay $60 for a pasture lot; A 
puts in 5 horses; B 4 horses; and C 3 horses ; how eli 
should each pay ? 


CHAPTER XIV. 
INVOLUTION AND EVOLUTION. 


240. Table for Determining the Number of Terms 
(Orders) in the Square Root of a Given Square 


Number. 
121 From the marginal table it is seen | 
ihre ate: that a square consisting of one or two | 
102—100 terms contains one term in its square 


O9F aU root; a square of three or four terms 
100710000 contains two terms in its square root; 
9997=998001 a square of five or six terms contains 


100021000000 three terms in its square root, etc. 
99992—99980001 


eae The principle may be formulated 
Ot On thus: 
.012=.0001 If a square number expressed in the 


.99?=.9801 — decimal scale be separated into periods of 
.0017==.000001 two terms each, beginning at the interval 
-999?==.998001 between units and tenths, there are as 


etc. many terms in its square root as there are 
periods in the square. 


INVOLUTION, 1QI 


241. Table Indicating the Lowest Place which a Sig- 
nificant Figure can Occupy in the written Product of 
the Given Factors. 


Remark. The following abbreviations are used—u for units, 
t for tens, h for hundreds, th for thousands, tth for ten thou- 
sands, .t for tenths, .h for hundredths, .th for thousandths, .tth 
for ten thonsandths, etc. 


{0 inte BD Coe Ub WO bese 1G 
12== h. Ho) 0 a,b, ie eee 
h?== tth. pn 0 eRe allte § Uke ag RG Tex iatiy ob 

th?—= m. Ti ae Geren LAD, GS hi ieeeth 
ace nh, Tht) te eatth: ath ==. ttn 
.h?2=— .tth. etc etc. 

etc. 


242. A number Consisting of Tens and Units Involved 
to the Second Power. 


Examiple. 347— what? 


Remark. In the light of the definition of a second power, 34 
is squared by multiplying it by itself. In order that the par- 
tial products be readily seen, they should not be combined by 
addition until they are all found. 

The involution may be shown in the form that follows. 


o4 
34 
16 = 42 = u?, 
Pees OU 4) = t KO 
ee a OO l= tC U1 
900 = 30? = t?. 
1156 = 342 = t? 4+ 2tx ust u?. 
Remark. Tf any number consisting of tens and units be 
squared, it is readily seen that the same steps will be taken as 
in the above example; i. e., the square of a number consisting 


of tens and units is composed of the square of the tens plus 
twice the tens multiplied by the units plus the square of the 


units. 


} 2 xu 


192 INVOLUTION AND EVOLUTION, 


243. A number Consisting of Hundreds, Tens and 
Units Involved to the Second Power. 
Example. 2342 = what? 

234 
234 
16 = 42 = v2, 
120 = 20:4 == Cay 
800 = 200 X 4=h xX u. 
120 "43650 oer 
900 = 802 = t?. 
6000) 200: x 30'=='hx 
800 34264200 == i ya 
6000 —=330" %, 260) == he cas 
40000 = 2002 == h?2. 


54756 = 2342 = h? +2hxt+t?4+2[h4+t]xutut. 


Remark. If any number consisting of h, t and u be squared 
the same steps will be taken as in the above example; i. e., the 
square of a number consisting of h, t and u is composed of the 
square of the h plus twice the h multiplied by the t plus the 
square of the t plus twice the sum of the h and t multiplied by 
the u plus the square of the u. 


244. ANumber Consisting of Tenths and Hundredths 
Involved to the Second Power. 
Example. .242 = what? 


24 
24 


0016 = .042 = .h?, 


(OOS tee 2s Sed eee bea 
008: == (04's aD 
Ohio ean ae 


0576 = 242° .t?-49:t« h-b ch, 


Remark, It is thus seen that the square of a number consist- 
ing of tenths and hundredths is composed of the square of the 
tenths plus twice the tenths multiplied by the hundredths gi 
the square of the hundredths. 


INVOLUTION, 193. 


245. ANumber Consisting of Tenths, Hundredths and 
Thousandths Involved to the Second Power. 


Example. .2467=what ? 
.246 
246 

.000036—.0062=.th?. 

.00024 =.04>.006=.hX.th. 

ato ==. .006=—.tX .th. 

00024 =.006x .04==.h X.th. 

016° .==.042—.h?. ; 

008 =.2x.04=.tX.h. 

0012, ==,006 .2—.t X.th. 

pow 04>. 2——.0>< hb. 

.O4 - 

.060516=.2462—.t?+2.t«.h+.h?2+2/.t+.h]x.th-+.th? 

Remark. It is thus seen that the square of a number consist- 
ing of tenths, hundredths and thousandths is composed of the 
square of the tenths plus twice the tenths multiplied by the 
hundredths, plus the square of the hundredths, plus twice the 
sum of the tenths and hundredths multiplied by the thou- 
sandths, plus the square of the thousandths. 

246. ANumber Consisting of Tens, Units and Tenths, 
Involved to the Second Power. 
Example. 32.62==what? 

32.6 
32.6 
~ 36==.62=.t?, 

1.2 == 2X.6==u X.t. 
Pe aa OU Oat ><. f. 
Uetgeen U6 Ae UK UL 

ee et TS 
60... = 80K2=tXu. 
ie 0 oUt x 
ee 2 OU== bX Uh. 
900.: 3) 142, 
1062. 1632. B= xan +t u) ae and t?3 


194 YNVOLUTION AND EVOLUTION, 


Remark. It is thus seen that the square of a number consist- 
ing of tens, units and tenths is composed of the square of the tens, 
plus twice the tens multiplied by the units, plus the square of 
the units, plus twice the sum ofthe tens and units multiplied 
by the tenths, plus the square of the tenths 


247. General Formula Embodying the Involution ot any 
Number in the Decimal Scale to the Second Power. 


A general principle for exhibiting the elements 
which compose the second power of a number consist-. 
ing of any number of terms (orders) in the decimal 
scale may be thus stated :— 


The square of a number consisting of any number 
of terms is composed of the square of the first, or high- 
est term plus twice the first term multiplied by the sec- 
ond, plus the square of the second, plus twice the sum 
of the first two terms multiplied by the third, plus the 
square of the third, plus twice the sum of the first 
three terms multiplied by the fourth, plus the square 
of the fourth, plus twice the sum of the first four terms — 
multipled by the fifth, plus the square of the fifth, 
plus, etc. 


248. Evolution of the Second Root. 
Example. 1. 7/1156 = what? 


Blackboard form. Thought form. 
1156(84 
ual Since there are four terms in the square 
6t)256 there are two terms in its square root, viz.— 
240 tens and units. 
16 The square of a number consisting of t 


16 and u is composed of the t?-++-2txu+u?. 
The sa. of th; hence the h of the power con- 


INVOLUTION, 195 


tain the sq. of the t of the root. The greatest square 
number of h in 11h is 9h, the sq. root of which is 3t. 
9h taken from the power leave 256. This remainder 
contains a product of which twice the tens of the root- 
or 6t, is one factor and the units of the root is the 
other factor. Since txu=t, the 25t of the remain. 
ing part of the power contain the required product 
Since 25t contain a product of which one factor is 6t, 
the other factor is found by dividing 25t by 6t; the 
quotient, 4, is supposed to be the units of the root. 
The product of 6t by 4 = 24t, which taken from the 
remaining part of the power leave 16. This remainder 
must contain the square of the u of the root. The 
square of 4= 16, which taken from the remaining part 
of the power leave nothing. 1156 is thus found to be 
a square number of which 84 is the square root. 


Example 2. 7/54756 = what? 


Blackboard form. Lhought form. 

54756(234 
4 Since there are five terms in the 
{40BR iven square there are three terms 
ey * iG a its square root,*viz.—h, t and u. 
Saar The square of a number consist- 
2756 ing of h, t and u, is composed of 
pee it the square of the h, plus twice the 
46t) 1856 h multiplied by the t, plus the 
184 square of the t, plus twice the sum 
16 of the h and t multiplied by the 

16 u, plus the square of the u. 


The sq. of h = tth; hence the tth of the power con- 
tain the sq. of the h of the root. The greatest square 
number of tth in 5tth is 4tth ; the square root of which 
is 2h. 4tth taken from the power leave 14756. This 
remairger contains the product of two factors, one 


196 INVOLUTION AND EVOLUTION, 


of which is twice theh of the root, or 4h, and 
the other the t of the root. Since hxXt =th, the 
th of the remaining part of the power contain the re- 
quired product .... Since 14th contain a product of 
which one factor is 4h, the other factor is found by di- 
viding 14th by 4h, the quotient, 3t, is supposed to be 
the tens of the root. The product of 4h by 3t = 12th, 
which, taken from the remaining part of the powet 
leave 2756. This remainder contains the sq. of the t 
of the root. The square of 3t = 9h, which taken from 
the remaining part of the power leave 1856. This 
remainder contains the product of two factors, one of 
which is twice the sum of the h and t of the root, or 
46 tens, and the other the units of the root. 


Since t X u = t, the tens of the remaining part of 
the power contain the required product. 


Since 185t contain a product of which one factor 
is 46t, the other factor may be found by dividing 185 
tens by 46 tens, the quotient, 4, is supposed to be the 
u of the root. The product of 46t by 4 = 184t, which 
taken from the remaining part of the power leave 16. 
This remainder must contain the square of the u of 
the root. The sq. of 4 = 16, which taken from the re- 
maining part of the power leave nothing. 


54756 is thus found to be a square number the 
square root of which is 284. 


Remark. It is observed that the tens and units of the root as 
found are supposed to be the correct numbers for those orders, 
respectively. Sometimes the obtained quotient may be so great 
‘that its product by the divisor will, when subtracted from the 
remaining part of the power, leave a remainder too small to 
contain the partial products that are yet to be taken out. Ifa 
quotient is too great a less quotient must be used. 


INVOLUTION, | 197 


Exercises. 


Evolve the second root of— 


1, 4096. - 14, 516961, 
2. 7396. 15. 104976. 
d. 5476. 16. 36372961. 
4, 7744. 17. 595984, 
5. 3721. 18. 492804. 
6. 9216. 19. 334084, 
7. 6889. 20, 21316. 

8. 763876, 21. 195364, 
9. 98596. 22. 2047761. 
10. 426409, 23. 120409. 
11. 662596. 24. 15376. 
12. 182329. 29. 222784. 
13. 6375625. 26. 67305616. 


249. Table for Determining the Number of Terms in 
the Cube Root of 2 Given Cube Number. 


eM From the table it is seen that a 
9°==729 cube number consisting of three terms 
103=1000 or less, contains one term in its cube 
998970299 root; that a cube number consisting 


100%—1000000 of four, five or six terms contains two 
999%==997002999 terms in its cube root; that a cube 

number consisting of seven, eight, or 
nine terms contains three terms in its cube root; ete. 


250. Principle. There are as many terms in the cube 
root of a cube number as there are periods of three 
terms each in the number, counting from the interval 
between units’ and tenths’ orders. 


Remarks. The highest period in a cube integer may contain 
but one or two terms. 

Z. The terms of the highest period of a cube decimal frac- 
tion may be wholly or partly represented by zeros. 

3. lhe lowest period of a cube integer may be wholly or partly 
represented by‘zergs Every period in a cube number may be 
partly represented By zeros. 


198 INVOLUTION AND EVOLUTION, 


251. Table Indicating the Lewest Place in Which a 
Significant Figure Can Occur in the Written Cube 
of Numbers in the Orders Named. 

Sige ea .t? = .th. — [For other indicated pro- 

sited) -h? -=.m. ducts see corresponding ta- 

he Sim, th? =.b. ble, page 191.] 

thea b wiiG. 

252. A Number Consisting of Tens and Units Involved 
to:the Third Power. 

Example. 384% = what? 
In the light of the definition of 8d power, a number 
is cubed by mutiplying its square by the number 
itself, — 


(30 + 4)? = 302+ 2 times 380 x 4+42= 1156 

30 +4 = 34 

302. 4-+ 2 times 80 x 42 + 48 4624 
303-1 2 times 802 «.4 + 30 x 42 3468 


303-L 8 times302 kK 4-+ 3 times 30 & 42 + 48 = 39304 
Hence.—The cube of a number consisting of tens and units is 
composed of the cube of the tens, lis 3 times the square of the 
tens multiplied by the units plus 3 times the tens multiplied by 
‘the square of the units, plus the cube of the uaits. 


-253. A Number Consisting of Hundreds, Tens and 
Units, Involved to the Third Power. 


Example. 2343 =: what? 

234 = 200 + 30 -+ 4. 

(200 + 30 + 4)? = 2002 + 2 times 200 x 30 + 
00? -+- 2 times (200 + 30) x 4+ 4? [See Art. 248.] 


Re-writing this formula after removing the parentheses we 
have— 


INVOLUTION. 3 199 


200?+-2 times 200 39-+-30?--2 times 200 4-2 times 3041-4" 


Multiply this formula by................ 200+-30-+-4 
and write the partial product below. eure AS 

Upon comparing these 2 times 80 x 4? 
partial products we find that 2 times 200 x 4? 
we have 2003 + 8 times 2002 BOT Ke 4 
<x 30+ 8 times 200 x 302 2 times 200 x 30x 4 
+ 30% + 3 times 200? « 4 JOOt <1 4. 
+ 6 times 200 * 80 x 4+ DO > 43 
3 times 30? <4+3 times 200 2 times 30? x 4 
x42-+3 times 30 x 42-43, 2 times 200 x 80x 4 

Factoring the 5th, 6th 30° 
and 7th terms of this formu- 2 times 200 « 30? 
la reduces them to 38 times 200? x 380 
(2002 + 2 times 200 x 30 200 x« 4? 
-+ 302) x 4. Observing that 2 times 200 x 30x 4 
the parenthetical quantity 2 times 2007 X 4 
equals (200 + 30)?, the re- te DUN Gs or 
duced expression becomes 3 2 times 200? « 30 
times (200 + 30)? x 4. 200° 


Factoring the 8th and 9th terms of the formula above, 
reduces them to 3 times (200 + 30) x 4?. 

Re-writing the formula, substituting 3 times (200 + 
30)? x 4 for the Sth, 6th and 7th terms, and 3 times 
(200-++-380) x 4? for the 8th and 9th terms, we have as 
the completed formula— 200% + 3 times 200? & 30 + 
3 times 200 & 30? + 303 + 3 times (200 + 30)? x 4+ 
3 times (200 + 30) x 4? +43. 

Any number consisting of h, t and u may be cubed 
in the same manner as the above; hence, the cube of 
a number consisting of hundreus, tens and units is 
composed of the cube of the hundreds, plus 3 times 
the square of the hundreds multiplied by the tens, 


200 INVOLUTION AND EVOLUTION, 


plus 3 times the hundreds multiplied by the square of 
the tens, plus the cube of the tens, plus 3 times the 
square of the sum of the hundreds and tens multiplied 
by the units, plus 3 times the sum of the hundreds and 
tens multiplied by the square of the units, plus the 
cube of the units. 

Thiseformula is abbreviated thus: h*+ 8h’? x t 
+3hxXx?4+ ¢+3(h + tx u+3(h+t) X wW +0. 


The cube of a number consisting of th, h, t and u is 
composed of th?-+ 3th? x h + 3th x h?+ h*®+ 
8(th + h)?xX t+ 3(th +h) x #4+¢+4 3¢th+h+ 
t? xu+ 38th+h+t)xw4+u*% 


Remark. A careful study of the above forms will discover 
the law by which a iormula may be constructed for the cube of 
a number consisting of any number of orders in the decimal 
scale. 

Evolution of the Third Root. 


Example. 7103823 == what? 


Blackboard form. Thought form. 
103823 (47 Since there are six terms in the 
64 power, there are two terms in its cube 
4Sh)39828 root; viz.—t and u. 
336 The cube of a number consisting 
GC Ale of tens and units is composed of t?+ 
588 av «x ut 38t xX wt ww. . 
CBee The cube of tens is thousands, 
343 hence the th of the power contain 


the cube of the t of the root. The greatest cube num- . 
ber of th in 103th is 64th, the cube root of which i5 
4t. The power diminished by 64th = 398238, which 
contains a product of which one factor is 3 times the 
square of the t of the root and the other is the u of the » 


EYOLUTION. 201 


root. 3 times the square of 4t — 48h. Since the 
product of h by u —h, the hundreds of the remain- 
ing part of the power contain the required product. 
8¢8h + 48h = 7, which is supposed to be the u of 
the root. 7 times 48h = 336 h, which taken from the re- 
maining part of the power leave 6223. This remain- 
der contains the product of 3 times the tens of the 
root by the square of the u of the root. 12t « 49 
= 588t, which taken from the remaining part of the 
power leave 343. This remainder must contain the 
eube of the u of the root. The cube of 7 = 348, which 
taken from the remaining part of the power leave 
nothing. We thus find that 103823 is a cube number 
and that 47 is its cube root. 

Remarsk. 1. In the light of the foregoing form the observant 
pupil will readily evolve the third root of any cube number. 


2. For the application of cube root in determining lines see 
text books on Arithmetic. 


Exercises. 
1. 592704. 10. 110592. 
2. 15625. ELS IS 26004: 
3. 3800763. 12. 2985984. 
4. 704969. 18. 36926037. 
5. 970299. 14. 150568768. 
6. 24889. 15. 113890625. 
7. 373248. 16. 347.428927, 
8. 103828. 17. 10077696. 
9. 274.625. 18. 705919947254, 


~ 14 


* CHAPTER XV. 
TEST PROBLEMS. 


1. If a merchant mark his goods 25% above cost, and 
sell them at 25% below the marked price, does he gain 
or lose and at what rate % ? 

2. A load of hay weighs 15 cwt. 21 lb.; if 2 ewt. 11 
lb. be sold, what part of the load remains ? 

3. Multiply .025 by 2.5. 


4. A owned # of a store and sold to B ¢ of his share 
and to C 2 of his share; what part of it did he still 
own? 

5. Required the cost of § of a yard of cloth, if 3 of a 
yard cost $7%. 

6. The longitude of Washington is 76° 56’ west of 
London, what change would it be necessary to make 
in a time-piece in coming from London to Washington? 


7. At what rate will $380 in 7 yr. 3 mo. yield $165.30 
interest ? 

8. }=what % of 9? 

9. if 94 eggs weigh a pound, and a pound of eggs 
equala pound of steak as food, at what price per dozen 
must eggs be bought in place i steak at 22¢ per lb.? 

10. Whatis the weight of the air in a room 25 ft. by 
20 ft. by 12 ft., water weighing 770 times as much as 


air, anda cubic foot of water WIEBE 1000 oz., Avoir- 
dupots? 


PROBLEMS. | 203 


11. If lead is 11.445 times as heavy as water, what 
is the weight of a piece of lead i m. by 2 dm. by 5 
cm.? 

12. A lot of goods was marked 40% above cost; if 
sold at 30% less than the marked price, was there a 
gain or loss and at what rate per cent.? 

13. Which would yield the better pay, a7% bond 
at 115 or a 6% bond at 98, and at what rate better? 

14> If a merchant sell flour at $9 per barrel, and wait 
6 months for his pay, at what price could he afford to 
sell for cash if money is worth 2% a month? 


15. For what sum must a 3 months’ note be drawn 
so that when discounted by a bank at 7-per cent. I 
may get $400? 

16. At what quotation must I buy a 6% stock to 
make as good an investment as froma 4% stock at 
70 ? 

17. A traveling salesman is allowed 12% commis- 
sion on his sales; his employer’s rate of profit is 20% 
on the goods sold ; what is the first cost of goods which 
the salesman sells for $7.66? 

18. A water tank is 3 ft. deep, 4 ft. long, and 4 ft. 
wide; it is supplied from a flat roof 20 ft. by 30 ft.; 
what depth of rain must fall to fill the tank ? 

19. Required the present worth of $1320 due in 3 yr. 
4 mo. without interest, if muney is worth 6%. 

20. A man bought a horse for $72 and sold it for 25 
per cent. more than it cost and 10% less than the ask- 
ing price; what did he ask for the horse ? 

21. 2 ft. 9 in. = what part of a rod? 

22. Cincinnati is 7° 49’ west of Baltimore; when it 
is noon at Baltimore, what time is it at Cincinnati? 


204 PROBLEMS. 


23. A man bought stock at 25% below par, and sold 
it at 25% above par; required his rate of gain. 


24 How many yards of carpet yd. wide will cover 
a floor i8 ft. by 15 ft. ? 

25. ¢ + %=—what? 

26. Multiply 15.04 by 1.6. 


27. How find the area of a circle if only the radius 
is given? Solve a problem. 


28. If a wheel turn 17° 30’ in 35 min. in what time 
will it make a revolution ? 


29. If rosin be melted with 20% of its weight of tal- 
low, what % of the weight of the mixture is tallow ? 


30. Required the weight inkilograms ofa bar of 
iron 2.6m. long, 6cm. wide, and 2 cm. thick, if 
iron is 7.8 times as heavy as water. 


31. With gold at 103 what rate of interest is made 
on a $1000 5-20 bond bought at 106? 


32. A and B are partners for 1 yr., A putting in 
$2000 and B #800: how much more must B put in at 
the end of 6 months to receive one-half the profits? 

33. How many grams does a DI. of water weigh? 

34.. Demonstrate the principle:—The product of the 
means = the product of the extremes of a proportion. 


80, The surface of the earth contains about 144000- 
000 sq. mi. of water, and about 53000000 sq. mi. of 
land. What. % of its surface is water? _ 

36. 3x F fers = what? 


. If a watch sell for $60 at a loss of 22 fo; fis what — 
should it sell to gain 30 per cent.? 


PROBLEMS. 205 


38. A broker bought stock at 8% premium and sold 
it at 9% discount thereby losing $510; how many 
shares did he buy? 

39. What is the net tax in a town whose taxable 
property is assessed at $430000, at 12 mills per dollar, 
5 per cent. being paid for collection ? | 

40. The difference of time between two places is 2 
hr. 15 min. 10 sec.; required their difference in longi- 
tude. 

41. Reduce .037 lb. Av. to drams. 

42. Reduce 84.5 ars. to square meters. 

43. An agent received $484.50 with which to buy 
sheep after deducting his commission at 2%; how 
much money did he spend for sheep? 

44. A policy for $2675 cost $53.30; find the rate of 
insurance. 

45. The difference of time between two places is 45 
min. 30 sec., and the place having the earlier time is 
in.longitude 85° 40’ west; required the longitude of 
the other place. 

46. Reduce .096 of a bu. to the decimal of a pint. 

47. A note of $125 dated May 3, 1883, and payable 
in sixty days, with interest at 5%, was discounted 
June 18, 1883, at 10% : required the proceeds. 

48. How many bu. of wheat will fill a bin 8 ft. by 5 
ft. by 4 ft.? 

49. How many meters of carpet .7 m. wide will 
cover a floor 4 m. by 4.5 m? 


50. Add 4 3 and 4. 

51. Multiply ? by 2.3. 

52. Divide 2.3 by 3. 

53. #— what part of 2.3? What % of 2.3? 


206 PROBLEMS. 


54. Reduce 135 sq. rd. 54 sq. ft. to the decimal of 
an acre. 

60. How much money must I remit to my agent 
to buy goods and pay himself $24 commiss!on at 14%? 

56. A sold a horse to B and gained 4 of its cost; 
B sold it for $80 and lost 4 of what it cost him; how 
much did A pay for the horse? 

57. Divide 3.45 by 1.5. 

58. A garden contains 800 sq. rd. and is 334 rd. long; 
how wide is it? 

59. What per cent. of 4 is $? 

60. What is 4 per cent. of .5? 

61. 428 is 7% more than what number? 

62. A man owes $300 due in 4 mo., $600 due in 5 
mo., and $100 due in 6 mo.; if he paid 4 of his indebt- 
edness in 2 mo., when in equity should he pay the 
balance? | 

63. What is the difference between the true and the 
bank discount of $859.50 for 90 days without grace, 
money being worth 8% ? 

64. A board is 20 ft. long and 9 in. wide; what is it 
worth at $30 per M? 

65. At what time between 6 and 7 o’clock are the 
hour and minute hands of a watch together? 

66. A broker bought 60 shares of stock at 1064, 
received a 5% dividend and then sold at 104; did he 
gain or lose and how much? 


67. How many ft., board measure, each board being 
18 inches wide, can be cut from a squared log 16 ft. 
long, 18 in. wide and 10 in. thick, allowing 4 in. for 
each cut of the saw? 3 


ete Se alia es AEN 
ee taht Pals ey 2 eee 
f aa 


PROBLEMS. 207 


68. If 12 men can ‘do a piece of work in 54 days, in’ 
how many days can 8 men and 5 boys do it, 1 man 
doing the work of 24 boys? 

69. If a note of $5000, for 4 mo., at 6% int. per an- 
nui, be discounted in bank, on any of making, at 8% 
per annum, what will be hy proceeds ? 

70. If it cost $312 to fence a field 216 rd. by 24 rods, 
what cost the fence of a sq. field of equal area ? 

71. A merchant bought goods at 20 cents per yd. 
and sold them at 40% profit, after allowing his cus- 
tomers 123% discount off; what was the marked price? 

72. A vessel, at noon, sails due north; after a cer- 
tain time an observation shows the sun to have sunk 
toward the west 2 signs 15 degrees; how long has the 
vessel been sailing? 

73. I sell a bill on London for £1675, “ the rate of 
24.3 cents per shilling: how much do I receive? 

74. If $6000 of 6% stock be sold at 90, and the pro- 
ceeds invested in 10% stock at 105, what will be the 
change in the income? 

75. I buy $1500 worth of goods at 4 mo., $850 at 3 
mo., $1750 at 5 mo.; what is the equated time for the 
payment of the whole? 

76. The square root of .1869 plus the square root of 
1296 equals what ? 

77. A owes B $1800; B offers to allow 5% off for 
cash; A pays $1425, how much is still due? 

78. Evolve the second root of 15625. 

79. What is the surface of a cube which contains 8 
times the volume of a cube whose edge is + of a foot? 

80. What is the capacity of a cylinder 20 ft. long, 
whose radius is 2 ft.? 


208 , PROBLEMS. 


81. I bought. goods in Europe, paid 20% duties, a 
commission of 2% upon duties and cost, and sold them 
at $10 per yard, clearing 34% on invoice price; re- 
quired the entire cost per yard. 

82. A travels 54 hours at the rate of 6 miles per hr., 
B then follows from the same point at the rate of 9 mi. 
per hr. ; how long will it take B to overtake A ? 

83. At 24.2 cents per shilling, what cost £1050? 

84. What principal in 3 yr. 4 mo. 24 da. at 5% will 
amount to $761.44 ? 


85. For what sum must a note dated April 5, 1886, 


for 90 da., interest 6%, be drawn, that when discounted 
at 7%, April 21, 1886, the proceeds may be $650 ? 


86. A room is 26 ft. long, 16 ft. wide and 12 ft. high; 
what is the distance from one of the lower corners diag- 
onally to the opposite upper corner ? 


87. St. Petersburg is 30° 19’ east longitude, and 
Indianapolis is 86° 5’ west longitude; when it is 3a. 
m. at St. Petersburg, what is Indianapolis time? 

88. Reduce 492 dekagrams to quintals. 

89. How many bricks, each 8 in. long, 4 in. wide 
and 24 in. thick, will be required for a wall 120 ft. long 
8 ft. high and 1 ft. 4 in. thick, no allowance being made 
for mortar ? 

90. If 6 men can build a wall 20 ft. long, 6 ft. high 
and 4 ft. thick in 16 days, in what time can 24 men 
build a wall 200 ft. long, 8 ft. high and 4 ft. thick? 


91. A man bought a square farm containing 140 
acres 100 sq. rd.; required the length of oneside. 

92. Evolve the 3d root of 592704. Of 2985984. 

93. A farmer exchanged 100 bu. of wheat at $1.25 
per bu. for corn at $.374 per bu.; how many bu. of corn 
did he receive? 


” as 
\ 
\ 

\ 


PROBLEMS, 3 209 


94. The sum of two numbers is 785: their differecne 
is 27; what are the numbers? 


95. Divide 498 by +4. 
96. Reduce .%; to a decimal fraction. 


97. How many loads are contained in a pile of wood 
40.16 ft. long, 7.04 ft. high and 4 ft. wide, if each load 
contains 1} cords? 


98. If I exchange $12000 of 8% stock at 115 for 
5% stock at 69, do I gain or lose on annual income, 
and how much? 

99. How large asight draft can be bought for $259.52, 
exchange being 12% premium? 

100. What are the cubical contents of a cylinder that 


_ will just enclose a sphere 9 in. in diameter ? 


101. Evolve the second root of 3%, to two decimal 
places. 


102. Three farms contain respectively, 356, 898, and 
1254 acres, which I desire to cut into building lots of 
the largest equal size possible; how many acres will 
each lot contain ? 


103. Divide 4 by 4. 
104. Multiply .303 by .08. 


105. The sun at 12 o’clock is over the meridian at 
Washington ; over what meridian will it be after trav- 
eling through 5 signs 5 degrees? What time will it 
then be at Washington? 


106. How many grams does a liter of rain water 


weigh? 


107. At 7 cents per square foot, required the cost of a 
brick walk 6 feet wide around a lot 200 ft. by 300 ft. 


} 108. What sum of money loaned at 6% for 10 mo. 


210 PROBLEMS. 


will yield as much interest as $750 loaned at 4% for 11 
months? 

109. Required the area of a circle whose radius is 10 
feet ? 

110. A, Band C eat 8 loaves of bread of which A fur- 
nishes 3, ce B5. C pays A and B 8 pieces of money 
of equal value ; how should they divide the money? 

111. For what sum must I make a bank note for 60 
days, which discounted at 10% will pay a $1000 debt 
now due? 

112. A street 60 ft. wide is crossed at right angles by 
another 80 ft. wide, what is the distance between diag- 
onal corners ? : 

113. A wall is 83 meters long, 2 Dm, high, and 5 dm. 
thick; what is its value at $3.30 per cu. meter ? 

114. a. What is the value of the N. E. 4 of the N. 
W. + of section 16, at $)2 per acre? 

b. Make a plat of the congressional township and 
indicate the part described. 

115. A miller takes for toll 4 quarts from every 5 bu. 
of grain ; what per cent. does he get? | 

116. In what time will $375.40 yield $37.54 interest 
at 6% per annum? 

117. A barn is 40 ft. wide; the comb is 165 ft. from 
the plate and the rafters are Of equal length ; what i 18 
the length of each rafter ? 

118. If money is worth 12%, what is thee true dis- 
count of $235.10; due one year hence ? 

119. In a cube whose edge is #in., how many cubes 
each + of an inch wide? 

120. A rectangular field 15 rods wide co7tains 3 
acres ; how long is it? 


PROBLEMS. 211 


121. How many yards of carpet 27 inches wide will 
cover a floor 18 ft. long by 14 ft. wide? 


122. If an article be sold for twice its cost, what 
is the rate per cent. of gain? Make and solve a prob- 
~ lem. ; 

123. If an article be sold for one-half its cost, what 
is the rate per cent. of loss? Make and solve a prob- 
lem. 

124. If an article be sold for one-third its cost, what 
is the rate per cent. of loss? Make and solve a prob- 
lem. 

125. Property worth $8760 is rented for $650 per 
annum; what rate of interest does the investment 
yield ? 

126. In what time will $2450 yield $725 int. at8%? 

127. A man receives $280 interest, annually, ona 
7% \oan ; what is the face of his loan? 


128. What principal will amount to $5750 in 3 yr. 5 
mo, 17 da. at 6% ? 

129. For what sum must property worth $6000 be 
insured at 5% to cover the premium and } of the 
property ? 

130. What is the selling price of corn whose first 
cost is 40 cts. per bu., freight 8% and rate of gain 
168% ? ; 

131. A merchantso!d two bills of goods for $50 each ; 
on one he gained 15%, and on the other he lost 15% ; 
required his gain or loss. 

132. What is the rate of interest on an investment 
in U.S. 4 per cents. at 95? 

133. A merchant is offered goods for $2500 cash, or 


wile PROBLEMS. 


for $2650 on 60 days time; which is the better offer if © 
money is worth 8% per annum ? 


134. For what must 5% stock be bought thatit may 
yield 7% interest on the investment ? : 

135. U.S.34’s bought at 98 pay what rate of interest 
on the investment? 

186. What sum of money will yield as much inter 
est in 10 months at 6%, as $1500 will yield in 12 mo. 
at 4% ? 

137. How much must be paid in U. 8. currency for 
a draft of £210 10s. 6d., exchange being 102, brokerage 
4%, and gold at 104? | 

138. A box is 4% m. long, 8 dm. wide and 3.5 dm. 
_ deep ; how many Keg. of distilled water will it hold ? 

139. A pile of wood is 15.5 m. long, 12 dm. wide, 
and 1.8 meters high; how many sters does it contain ? 

140. 3 = what % of $? 

141. Two men paid $150 for a horse; one paid $90 
and the other paid $60; they sold it so as to gain $75 ; 
what was the share of each ? 

142. A man owes A $105, B $75, and C $120; he has 
only $125. How much should he pay to each ? 

143. What is the face of a note payable in 90 days, 
~on which $3500 can be obtained at a bank, discounting 
at 6%? 

144, An irregular mass of metal immersed in a ves- 
sel full of water caused 2.25 1. of water to overflow the 
sides of the vessel; what was the weight of the metal 
if its specific oe was 7.2? 

145. What is the price, at $11.65 per kilogram, of 1 
liter of alcohol, its s. g being .79? 


PROBLEMS. | 213 


146. What is the capacity in liters, of a cylindrica] 
cup 16 cm. in diameter and 1.3 dm deep ? 

147. A and B gain in business $4160; A is to have 8% 
more than B, how much will each receive ? 

148. A merchant insures a cargo of goods for $3456 . 
at 35%, the policy covering both property and premium; 
what was the value of the property ? 

149. If a stack of hay 83 ft. high, weigh 7 cwt., wh: t 
weighs a similar stack 15 ft. high? 

150. For what sum must a note be drawn at 3 mo. 
that the proceeds, when discounted, without grace, at 
a bank, at 8% shall be $1274? 

151. The contents of a cubical block of stone are 
4913 cu.cm.; required its superficial contents in sq. m. 

152. I imported 12 casks of wine, each containing 
48 gallons invoiced at $2.75 per gal.; paid $108 for 
freight, and an Ad Valorem duty of 36% ; what is my 
rate of gain if I sell the whole for $3258? 


153. What is the area of a field whose parallel sides 
are 90 rods and 124 rods long, respectively, and the 
perpendicular distance between them is 50 rods? 

154. A has $550, and B has $330; what per cent. of | 
the money of each is the money of the other ? 

155. A farmer’s wagon loaded with wheat weighed 
4912 lb., and his wagon alone weighed 920 lb., what 
was his load of wheat worth at 95 cts. per bu. ? 

156. A man paid $30.09 for the use of $204 for 11 
mo. What was the rate of interest ? | 

157. What was due Jan. 1, 1886 on a note given Sept. 
1, 1884, the principal being $430, the rate 7%, per an- 
num ; endorsed Aug. 4, 1885, $34, July 7, 1886, $118? 


10 


15 


20 


CHAPTER XVI. 
Review Questions and Topics. 


Define mathematics.—What is conditional for 
extension ?—What branches of knowledge involve 
a study of extension?—What is known of every 
conscious mental state?—How are mental states 
known to be distinct?—How does the idea of 


number arise ?-- What is conditional for the num- 
erical idea ?— What attribute furnishes the basis of 
number in particular?—By what process does 
the mind obtain the idea one 2?— Define the zntegral 
unit, or unit one.—Define a fractional unit; a mul- 
tiple unit— What is the primary idea in Arith- 
metic ?—How are other units related to the wnt 
one ?—What is a unit object ?—Define a number. 
—Of what units may a number be composed ?— 
On what basis are numbers classified as integers 


and fractions ?—Define an integer; a fraction.— 
On what basis are numbers classified as abstract 
and concrete ?—-Define an abstract number ;.a con. 
crete number.—On what basis are abstract inte- 
gers classified as prime and composite ?— Define a 
composite number; a prime number.—On what 
basis are numbers classified as simple and de- 
nominate ?— Define a simple number ; a denomin- 
ate nuinber ; acompound number.—On what basis 
are nunnrbers classified as positive and negative >— 


30 


40 


50 


55 


REVIEW. 215 


Define a positive number; a negative number.— 
Define notation.—What kinds are usually pre- 
sented in Arithmetic?—What is the alphabet 
of the Roman notation ?—What does each letter 
signify ?—What are the limits of the Roman nota- 


tion?--State the principles.--What is the alphabet 
of the Arabic notation?— W hat is the signification of 
each character?-—How many and what distinct sys- 
tems of numbers make use of the Arabic characters? 
—Define a scale ; the decimal scale.—W hatisa unit 


of the first order ?— How are units of higher orders 
formed ?— What is a period in the decimal system 
of numbers ?--What orders are embraced by any 
period ?—What is meant by a decimal division of 
1 ?—How are lower orders of units formed ?—State 


the principle of the decimal scale.—What is the 
office of the decimal point ?—Describe units’ place. 
—How many and what kind of units may be 
expressed therein ?—Describe tens’ place.—How 
many and what kind of units may be expressed 


therein ?—How may higher decimal units be ex- 
pressed ?—How may lower decimal units be ex- 
pressed ?— W hat is the representative scale? What 
is the form value of a figure?—What is the place 
value of a figure?— W hat is it to read a number ?>— 
In reading anumber where should the word and be 
used?7Name ten consecutive units in the fractional 
scale. How is each unit in the fractional scale 
numerically related to the one below it?—How 
many and what numbers are necessary to the idea 
of a fraction ?—What are these numbers together 
called :—Define the denominator ; the numerator. 


210 REVIEW. 


60 


70 


79 


80 


85 


90 


—What is the denomination of a fraction ?— What 
exception ?— How is a fraction notated ?— How 
may a decimal fraction be notated?—In com- — 


pound numbers the number of denominations 
is how limited?—What kind of a scale has ~ 


each of the common measures ?—How is a com- 
pound number notated ?—How are the parts 
of a compound number written as to abbreviation 


and punctuation ?—How is a compound number 
read ?— Define reduction ; descending ; ascending ; 
and state how each is effected ?—What aids does 
the child necessarily use in first performing the’ 
number processes ?—Define sum, addition, ad- 


dends.—State the mental acts involved in addi- 
tion.—State the principles of addition.—Show 
that reduction may be involved in finding a sum. 
—Which reduction ?—Describe and state use of 
the sign of addition.—State and illustrate the 


principles of addition.—Define product; multiph- 
cation; multiplicand; multipker; factor.—State 
the genesis and nature of multiplication.—If the 
multipher is an integer what relation to addition 
may be observed ?—If the multiplier is negative 


how does the product. compare with that produced 
by a positive multiplier ?—If the multiplier is of 
units’ order, of what order is the product ?—-If the 
multiplier is of units higher than of units’ order, 
of what order is the product ?—If the multiplier 
is of units lower than of units’ order, of what or- 
der is the product ?—Show that reduction is often. 
involved in multiplication.— Which reduction ?— 


Describe and state use of the sign of multiplica- 
tion.—State and illustrate each’ of the nine prin- 


95 


100 


105 


REVIEW. 217 


ciples given under multiplication.—How is the 
continued product of factors found?—Define a 
composite number; composition ; a prime num- 
ber.—Under what conditions are numbers rela- 
tively prime ?—Define a common multiple; the l. 


c. m.; a common factor; the greatest common fac- 
tor.—State and illustrate each of the vrinciples 
given under coraposition.—Define a power ; invo- 
lution; a root; second’ power; second root. — 
How are higher powers formed and named ?— 


How are roots named ?—What is the first power 
and first root of a number ?—Describe and define 
the index of a power.—Repeat.the table of squares 
given ; the table of cubes.—Under how many and 
what heads is the synthesis of numbers discussed? 


—How many and what methods of synthesis are 
there?—What relatior do the other processes 
called Synthetic sustain to the one method of syn- 
thesis ?—State in substance the last four remarks 


110 given under synthesis ?—Show the relation of sub- 


115 


120 


traction to addition.—Define difference, giving 
two definitions.—Define subtraction; minuend | 
subtrahend.—State the mental acts involved in’ 
subtraction.— Describe the sign of subtraction and 
state its use.—State and illustrate each of the 
principles of subtraction.—Under what condition 
must the minuend be prepared before a subtrac- 


tion is effected?—In what consists the prepara- 
tion ?--Which reduction is involved ‘-—How may 
reduction be avoided in subtraction ?>—Define 
quotient; division, dividend; divisor.—Define 
each of these terms in the light of its relation to 
multiplication.—What. is meant by a constant 
subtraction ?>—State how division grows out of 


15 


218 REVIEW. 


125 subtraction ?—State Prof. Olney’s position on the 


genesis of division.—How is division related to 
multiplication ?—In what consists the mental act 
of division ?—Define the quotient of one number 
by another; the dividend; the divisor.—Which 
130 reduction is sometimes involved in division ?— 


Illustrate.—If the divisor is of units’ order, of 
what order is the quotient ?—If the divisor is of a 
higher order than units’, of what order is the quo- 
tient ?—If the divisor is of lower order than units’, 
135 of what order is the quotient ?=State and illus- 


trate each of the principles of division ?—State 
and illustrate each of the six general principles of 
division.—Define disposition.—How is disposition 
related to composition ?—How are the factors of a 
i40 number found?—State and illustrate each of the 


principles of disposition.—Under what condition 
is a number known to be p,ime ?—Define evolu- 
tion ; the index of a root.—Describe and state the 
use of the radical sign ; exponent.—W hat is signi- 
145 fied by a fractional exponent other than a frac. 


tional unit °—Under how many and what heads 
is the analysis of numbers discussed ?—State the 
substance of the last four general remarks given 
under analysis.—Define g. c. d.—State and illus- 
150 trate the principles under this head.—Solve an 
example illustrative of each method given for 
finding g. c, d.—Define 1. c. m.—State the princi- 
ple and solve an example, using the form given 


on page 80.—If given numbers are not readily — 
155 factored how may their l. c. m. be found ?—Why? 


Define a fraction, giving two definitions.—Accord- 
ing to the primary idea of a fraction, what is the 


1 60 


165 


170 


180 


185 


REVIEW. 219 


reaximum value of a fraction ?—How is an ex- 
pression like $ to be interpreted ?— How is a frac- 
tion notated ?—On what basis are fractions classi- 


fied as decimal and common?—Define a decimal 
fraction.— How is a decimal fraction usually nota- 
ted?— How may it be notated ?—What is a com- 
plex decimal ?—Define a common fraction.—On 
what basis are fractions classified as proper and 


improper ?—Define a proper fraction ; an improper 


fraction.—State wherein there is no ground for 


this classification.—On what basis are fractions 
classified as simple, compound and complex ?— 
Define a simple fraction ; a compound fraction ; a 


complex fraction.—Show wherein this classifica- 
is not well founded.—State the likeness of a frac- 
tion to division.—State the principles of multiph- 
cation of fractions; of division; of reduction ; of 
relation.—How may a fraction be reduced to its 
lowest terms ?—Under what condition is a com- 
mon fraction not reducible to a decimal ?—Under 
what principle is addition of fractions effected? 


‘Under what principle is subtraction of fractions 


effected ?—State the two cases under which multi- 


plication of fractions is presented.—Solve an ex- 
ample under each case.—State the two cases under 
which division of fractions is presented.—Solve an 
example under each case, using —a., common frac- 
tions ; 0., decimal fractions.—Show that division 


of fractions is the reverse of multiplication of 
fractions.—State table of Aliquots.—Into what 


_classes are compound numbers divided ?—On 


what basis?— Write the diagram.—State the or- 


190 der to be pursued in the study of a ‘ measure.”— 


220 


195 


200 


205 


210 


215 


220 


REVIEW. 


Recite the tables of compound numbers.—How 
many and what cases of reduction descending in 
compound numbers ?—Solve an example under 
each case.—How many and what cases of reduc- 
tion ascending in compound numbers ?—Solve an 


example under each case.—Define Longitude.— 
What m>ridian is generally taken as prime ?7-Make 
a tableof corresponding time and longitude units. 
What is meant by relative time?—Show that all 
places have not the same relative time.—Explain 


what is meant by “‘ Standard time.’”—Solve exam- 
ples in time and longitude.— W hat is area ?— W hat 
is a volume, or solid ’—Solve an example in which 

area is computed.—Solve an example in which ~ 
volume is computed.—Show that length and breadth 


are not multiplied together.—If they could be 
would the product be surface?—W hy ?—Name the 
three primary units of the decimal system of meas- 
ures.—Which of these is the fundamental unit of 
the system ?—How does the meter comparein length 


with the yard ?—Name the land unit.— What is its 
area?—By what units are most surfaces measured ? 
— What is the wood unit ?—What is its volume? 
By what units are most volumes measured?— What 
units are used in measuring great weights ?—How 
does each of these compare with the gram ?—Name 


each prefix designating a secondary unit.—Con- 
struct tables for the measures of length, surface, 
volume, capacity, and weight, in the decimal sys- 
tem.—What relations exist by means of which 
capacity and weight are readily found from exten- 
sion ? What is percentage ?—Name the essential 
terms of percentage.—Define each.—Define amount 
and difference, and state which of the essential 


225 


— 230 


— 235 


240 


245 


250 


255 


REVIEW. 221 


terms includes them.—State the relations of per- 


centage.—a. To multiplication.—b. To factoring. 
c. To fractions.—State each of the gereral cases 
of percentage.—Give the solution of eash.—How 
many and what elements are involved in the first 
class of applications of percentage ?°—Name the 


principal applications of the first class.— What ele- 
ments are involved in the second class of applica- 
tions of percentage ?—Name the principal applica. 
tions of the second TON anite profit and loss; 
cost ; selling price; gain; loss.—State the relation 


of profit and loss to percentage by naming the cor- 
responding terms.—W hat is an agent ?—A factor? 
—A broker ?—A commission merchant ?—Define 
commission; brokerage. — State the relation of 
commission and brokerage to percentage by nam- 


ing the corresponding terms.—Define stock as used 
in percentage ; a corporation ; a firm; a company ; 
a share; a certificate of stock; a dividend; par 
value ; market value ; premium; discount.— How 
is stock quoted ?—State the relation of stock to 


percentage by naming the corresponding terms. 
—-Define insurance; fire insurance ; marine insur- 
ance; life insurance ; policy ; face of policy ; policy- 
holder ; underwriter ; premium.—State the relation 
of insurance to percentage by naming the corre- 
sponding terms.—Define a tax; a property tax; a 
poll tax ; an income tax; an excise tax; an assess- 
or; his duties; an invoice; tare; leakage; break- 
age; gross weight; net weight; specific duty: 
ad valorem duty.— Define interest; principal; 
amount; rate; legal rate.—What is the time unit 
in interest ?—Define simple interest; a partial pay- 


22:2 


* 


260 


265 


270 


275 


285 


290 


REVIEW. 


ment.-~ State the United States rule for computing 
interest in partial payments; the merchants’ rule. 
—What elements are involved in simple inter- 


est ?—What relations are sustained. among the 
elements ?—Define compound interest ; annual in- 
terest; discount; true discount; present worth.— 
State the relation of true discount to simple inter- 
est by naming the corresponding terms.-—Define 


bank discount; term of discount; proceeds ; days 
of grace; face of a note; legal maturity ; a protest- 
—.i{ an interest bearing note be discounted in 
bank, on what is the discount computed ?—State 
the relation of bank discount to simple interest by 


naming the corresponding terms.— Define commer- 
cial discount.—State its relation to percentage.— 
Define exchange; a draft; a sight draft; a time 
draft; the face of a draft; the course of exchange. 
—Who is the drawer or maker of a draft?—The 


drawee?—The payee ?—State the relation of ex. 
change to simple interest by naming the corre- 
sponding terms.— What is meant by the equation 
or average of payments?—Show the relation of 
equation of payments to simple interest.—Define 


Ratio.—How many and what terms are used in 
thinking a ratio?—Define each.—How is a ratio 
notated ?—Definea simple ratio; a compound ratio; 
an invorse ratio.—State each of the six principles — 
of ratio.—Define proportion ; asimple proportion ; 


a compound proportion.—How is a proportion 
notated?-—State and demonstrate the principle of 
proportion.—Under what condition is a number 
divided into proportional parts? State the prin- 


ciples. 


APPENDIX. : 


To Find The True Remainder. 


The following is a method for determining the true 
remainder. | 
Example. Divide 1377 by 294, using the prime fac- 
tors of the divisor, and determine the true remainder. 
Solution. The prime factors of 294 are 2, 3, 7, 7. 


2)1377 Hee aiass Benge 
MEIN 21250) Vg 2 re ae og 6 wie oi eck irs oc’. 1 
CS OAs a ae 6 oe oP Pa eh ae ORR 2 
1 | Ry al rae aS ee ae eS e ofosuns te) 
20 Gee : ye nab A Waseak wo SOO 
PTEOAPOTNIBITICOL Oyu eo e's cig oe kw eas 201 
Explanation. 


a. 1. Theentire dividend divided by 2 gives a quo- 
tient of 688 and a remainder of 1. | 

2. 688, which is approximately 4 the entire divi- 
dend, divided by 3 gives a quotient of 229 and a re- 
mainder of 1. 

3. 229, which is approximately 4 of the entire divi- 
dend, divided by 7 gives a quotient of 32 and a re- 
mainder of 5. 

4. 32, which is approximately j, of the entire divi- 
dend, divided by 7 gives a quotient of 4 and a re 


mainder of 4. 
5. 4, the final quotient is approximately 4, of the 


entire dividend, or the part required. 


CLP, Maras APPENDIX. 


6. 1. If1 remain upon dividing 4 the dividend, 2 
times 1, or 2, wou'd remain upon dividing the entire 
dividend. 

2. If5 remain upon dividing 4 of the dividend, 6 
times 5, or 30, would remain upon dividing the entire 
dividend. 

3. If 4remain upon dividing #, of the dividend, 42 
times 4, or 168, would remain upon dividing the eu- 
tire dividend. 

4. We thus find that upon dividing the entire divi- 
dend we should have remaining 1+2+30+168, or 201, 
as the ultimate, or true remainder. 


The accuracy of this result may be tested | y divid 
ing the entire dividend by the divisor as a whole. 


Rules for Squaring Numbers. 


I. To Square anumbcr ending in d. 
1. Square the integer. , 
2. Add 4 the integer. 
3. Add +. 


Il. To Square a number ending in 3. 
{2 Square the integer. 
a 


‘ 


2. Add the integer. 
3. Add 4. 


b. Multiply the integer by the integer next greater 
and add 4. 


Ill. To Square a number ending in 3. 
1. Square the integer. 
2. Add 3 of the integer. 
3. Add +. 


APPENDIX. 


to 
to 
Ci 


IV. To square a number ending in 5. 


1. Square the tens. 
a< 2. Add the tens. 
3. Annex 25. 
b. Multiply the simple value of the tens by the 
number next greater and annex 25. 


V. To square a number between 25 and 50. 
1. Take 25 from the number. 
2. Take the difference from 25. 


3. Square the remainder. 
4. Add the first difference as hundreds. 


VI. To square a number between 50 and 75. 
l. Take 50 from the number. 
2. Add the difference to 25. 
3. Call the result hundreds. 
4. Add the square of the difference. 


Vit. To square a number between 75 and 100. 
1. Take the number from 100. 
9. Take the difference from the number. 
3. Call the result hundreds. 
4. Add the square of the first difference, 


VIII. To square a number ending in 25. 
1. Square the hundreds. 
2. Add 4 the hundreds. 
3. Call the result ten-thousands., 
4, Add 625. 


[X. To squars a number ending in 75. 
1. Square the hundreds. 
2. Add 3 of the hundreds. 
8. Call the result ten-thousands. 
4, Add 5626. 


226 APPENDIX. 


CONTRACTIONS IN MULTIPLICATION. 


To multiply a number of two orders by 11. 
Think the sum of the terms of the multiplicand be- 
tween them: 


As—34X11=374, 64X11=594. 
To multiply together two numbers ending in 5, 
{a Find the product of the tens. 


2. Add $ the sum of the tens. 
3. Annex 25. 


To multiply together two numbers ending in $. 


1. Find the product of the integers. 
2. Add 4 the sum of the integers. 
3. Add 4. 


To multiply if one part of the multiplier is a multiple 
of the other. 
First Example. 
3645 =: the multiplicand. 3 

246 =the multiplier, its two digits, 24, repre- 
21870 = sentinga multiple of 6. 3645 x 6=21870. 
87480 Multiplying this by 4 (tens) we obtain 
896670 87480 (tens), which is 240 times 3645. The 


two partial products combined equal the 
required product, 896670. 
Second Example. 
784 = the multiplicand. 
856 = the multiplier,—its two digits 56, repre- 
627200 senting a multiple of 8. 627200 is the 
43904 product of 784 by 800. 44,5 of 627200 = 
671104 6272, which is the product of 784 by 8, 
and 7 times 6272 = 43904 which is the 
product of 784 by 56, Thetwo partial 
products combined equal the required 
product. 


ao 
Rules in Mensuration. 


| 
& 
| 


APPENDIX, | 227 


Exercises. 


1 UDI 7848 by 648. 
2 3674 ‘ 1224, 
3. a 4678 “ 729. 
4, i 9867-369... 
5 COOP Oro LI44: 
6 ¥ 6789‘ 730. 


To find the circumference of a circle. 


Multiply the diameter by 3.1416. 


To find the area of a crew. 
1. Multiply the square of the diameter by .7854; or 


_ 2. Multiply half the circumference by half the 
diameter. 


To find the diameter of a circle. 
1. Extract the square root of the area and multi- 


ply the result by 1.1284; or 


2. Divide the Rn Teen cs by 3.1416. 

To find one side of an inscribed equilateral triangle. 
Multiply the diameter by .86. 

To find the side of an inscribed square. 

1. Multiply the diameter by .7071; or 

2. Multiply the circumference by .225. 

To find lhe area of a circumscribed square. 

Square the diameter. 

To find the side of a square equivalent to a given circle. 
1. Multiply the diameter by .8862; or 

2. Multiply the circumference by .282. 


3228 ‘ APPENDIX. 


To find the area of a circular ring. 

Square the radius of the larger circumference; 

Square the radius of the smaller circumference ; 

Take the difference between these squares and 
multiply it by 3.1416. 

To find the area of an ellipse. 

1. Multiply the product of 4 the two axes by 
5.1416; or 

2. Mutiply the product of the two axes by .7854. 


To find the area of a sector of a circle. 


1. Multiply the arc of the sector by half the 
radius. 

2. Next find area of the circle. 
3. Now 360° : arc of sector :: area of circle : area 
of sector. . 


To find the area of a segment of a ctrele. 


1. Find area of sector having same arc. 

2. Find the area of the triangle formed by the 
arc of segment and the two radii. 

3. Subtract the area of triangle from area of sector. 

The remainder is the area of the segment if it be 
less than a semi-circle; or 

Add the area of the triangle to that of the sector 
if the segment is greater than a semi-circle. 


NOTES.ON THE ELEMENTS OF ALGEBRA. 


[In the brief notes on Algebra that are here given, 
no attempt at exhaustiveness has been made. Neither 
has a logical order of presentation been kept in view. 
It is hoped, however, that the definitions submitted 
will stimulate the student to think beyond the symbol 
to that which is symbolized, and that the methods of 
treating the few topics that are given will prove satis- 
factory and helpful to the beginner and to the teacher 
of beginners in Algebra. | 
I. Quantity is the amount or extent of that which 
may be measured or numbered. 

Remarks. 1. A quantity and a number are terms that usually 
embrace the same content in Algebra. 

2. Both known and unknown quantities are in- 
volved in Algebraic investigations. [Define each 
kind. | 
2. Equation. Two equal quantities viewed with ref- 
erence to their relation of equality constitute an 
equation. 

3. The Members of an equation are the two quanti- 
ties constituting the equation. 

4. The Transformation of an equation consists in so 
operating upon either or both members as to change 
some or all of the symbols in the expression ; or, in so 
operating upon both members as to change their value 
in the same ratio. 

5. Axioms of Transformation. 

1. If the same quantity (or equal quantities) be 
added to each member of an equation, the resulting 
sums are equal. 


230 APPENDIX. 


2. If the same quantity (or equal quantities) be 
subtracted from each member of an equation, the 
resulting differences are equal. 

3. If each member of an equation be multipled by 
the same quantity (or by equal quantities) the result- 
ing products are equal. : 

4, If each member of an equation be divided by the 
same quantity (or by equal quantities) the resulting 
quotients are equal. 

6. Ifeach member of an equation be involved to the 
same degree, the resulting powers are equal. 

6. If tae same root be evolved from each member of 
an equation, those roots are equal. 

7. If an operation indicated in the expression of 
either or both members of an equation be performed, 
the value of the member remains unchanged. 


6. Algebra is that branch of pure mathematics which 
treats of the nature and properties of the equation and 
of its use as an instrument for conducting mathemat- 
ical investigations. [Olney. ] 

Remark. In Algebra quantities are frequently represented 
by a literal notation ; the letters of the English alphabet being 
used. 


7. Algebraic Symbols. 
a. Of Quantity. 

1. Of known quantities—The Arabic symbols of 
number, and the first letters of the alphabet. 

2. Of unknown quantities—The last letters of the 
alphabet. | 

3. 0, called zero, represents a quantity infinitely 
amiallay' 

4, co, called infinity, represents a quantity infinitely 
great. “ss 


APPENDIX. 231 


b. Of Operation. 
1. Of Addition.—The sign, +-. 
2. Of Subtraction.—The sign, —. 


3. Of Multiplication.—The sign, x; the period 
placed between two factors, as @.6. Also two symbols 
of quantity (either or both being literal) written with 
no sign between them are considered as indicating the 
product of the quantities represented by the given 
symbols, as—ab indicates the product of the factors 
represented by a and 0. 


4. Of Division.—The sign, +. [Give others. | 
5. Of Involution.—A positive integral exponent. 
6. Of Evolution.—The sign, 7’, or a written frac- 
tional unit as an exponent. [See page 75. ] 
c. Of Relation. 
. Of equality.—The sign, =. 
. Of inequality.—The sign >, or <. 
. Of ratio.—The colon (:). 
. Of equality of ratios.—-The double colon, (::). 


m Ww bo 


5. Of aggregation.—The parenthesis, the bracket, 
the brace, the bar, and the vinculum. Co Be on Gee Nc 
, 
8. Algebraic Terms. 


Remark. An algebraic expression is a representation of a 
quantity by means of algebraic symbols. 


1. An algebraic term is a quantity not thought as 
related to another by addition or subtraction ; or, it is 
any one of the parts of a quantity whose parts are 
thought as related by addition or subtraction. 

2. A monomial, is a quantity consisting of but one 
term. 


232 APPENDIX. 


3. A polynomial is a quantity consisting of more 
than one term. 

Remark. A polynomial whose expression is affected by a 
sign of aggregation is sometimes called a compound term. 

4, A binomial is a quantity consisting of but two 
terms. 

5. A Trinomial isa quantity consisting of but three 
terms. 

6. The degree of a term is the ordinal of the number 
of literal factors in its expression. 

7. Similar terms are those whose expressions have 
the same literal factors throughout, the like literal 
factors having the same exponents. 

8. A coefficient is any factor of a term. 

Remark. The term coefficient is usually applied to the first 
factor of a term. 

9. A positive term is one whose expression is pre- 
ceded by the sign, + or by no sign. 

10. A negative term is one whose sees: is pre- 
ceded by the sign, — 

9. Addition. 

[Norse. For definitions and principles see pages 35 

to 38.] | | 
Remarks. 1. Addition may be defined as the process of combin- 
ing in the fewest possible terms the aggregate value of quanti- 


ties. 

2. The signs, -[- and —, as symbols of operation sig- 
nify addition and subtraction, respectively. They are also 
used to denote the relative character of a quantity. 

The use of these signs as symbols of character is often illus- 
trated as follows: If distance up is considered as positive, dis- 
tance down is negative. Tf direction to the right is considered 
as positive, direction to the left is negative. If a man’s gain be 
considered as positive, his loss is negative, &e. Each sign indi- 


APPENDIX. f 233 


cates the opposite or negative of the. other ; so that:if, a positive 
quantity be applied to a negative quantity of the same absolute 
or numerical value, the sum, or effect obtained is nothing. If 
-|-5 be applied to —3, the —3 negates or balances 3 units of 
the -|-5, giving as the sum, or effect of the combination, —- 2. 
Again, if --4 and —7be applied .to each other, ——4 negates’ 
or balances —4 of the —7, giving —3.as the sum, or‘effect of 
the combination It may thusbe said: a. That ane sum of two 
quantities whose absolute values are equal but whose charac. 
ters are unlike is nothing. 
b. That the sum of two quantities of unlike character ana of 
unequal absolute values is the difference between their abso-* 
lute, or numerical values, the character of that difference being: 
positive if the absolute value of the positive addend is the 
greater and negative if the absolute value of the negative ad- 
dend is the greater. [See page 18 ] ; 


. [ Exercises. ] 
10. Subtraction. 

1. In Arithmetic, subtraction is, perhaps, most fre- 
quently viewed as the act of separating the minuend 
into two parts, one of which (the given part) is called 
the substrahend and the other (the required part) the 
remainder. As thus consid2red, the process is essen- 
tially analytic. In Algebra the minuend is often nu- 
merically less than the subtrahend. In such cases, and > 
indeed in all cases, subtraction may be viewed synthet- 
ically—as the process of building up the minuend > 
(sum) by supplying from memory the quantity 
(whether it be positive or negative) which added to the 
subtrahend will make the minuend. 


This simple view of subtr..ction effectually prevents 
the difficulty that pupils so frequently meet with in | 
regard to the signs in subtraction. | 

2. In no case is it necessary for the sign of the subtra- 
hend to be changed in effecting a subtraction. 


16 


234 APPENDIX. 


3. In subtraction, the minuend, subtrahend and dif- 
ference are similar quantities. 

If, for example, we are to subtract 2a from 5a, it is 
easily done by thinking 3a as the quantity which 
must be added to 2a to make a sum equal to da. 

But if the problem be to subtract b from a, its solu- 
tion is more complex. In the light of Prin. I, page 
58, 6 cannot be subtracted from a since they represent 
dissimilar quantities. In order to explain the sub- 
traction by which the simplest expression for the re- 
quired difference. is found, 6 must appear in the ex- 
pression of the minuend and a must appear in the ex- 
pression of the subtrahend: thus— 


The minuend = a-+ Ob. 
The subtrahend = 0a + 6. 
The difference = a— 6. 


Now minus 6 must be added to 6 to make a sum 
equal to 0b; while a@ must be added to Oa to make a 
sum equal toa. The required difference may thus be 
expressed by a — 6. In any case, therefore, if a term 
occur in the minuend while a similar term does not 
occur in the subtrahend, the literal expression of such 
term with zero for a coefficient should be placed in the 
expression of the subtrahend: Or, if a term in the 
minuend be numerically expressed, while the corres- 
ponding term in the subtrahend is vacant, zero, alone, 
should be written in the vacant place. 

A like deficiency in the minuend should be sup- 
plied in the same manner. 

The subtraction can then be explained consistently 
with the definitions and principles of subtraction. 


[For a further discussion of subtraction see pages 
56 to 58.] 


APPENDIX. 235 


11. Multiplication. 
[For a discussion of the nature of multiplication, its 
definitions, principles, etc., see page 44, et. seq. ] 
THE SIGNS IN MULTIPLICATION. 


1. Multiply @ by 8. 
Solution. -*.: the multipher = b times 1, the pro- 
duct = 4 timesa =ab. .«.a xX b-=ad. 


2. Multiply — a by 4. 
Sqlution. °*.. the multiplier = 4 times 1, the pro- 
Auct —b times —a = — ad. .. —a xX 6 = — ab. 


3. Multiply @ by — 4. 

Solution. °*.* the multiplier = the negative of 4 
times 1, the product = the negative of 6b times a, 
6 times a = ab, the negative of which is — ab. 

aX —b = —ab. 

4, Multiply — a by — 6. 

Solution. °.. the multiplier = the negative of b 
times 1, the product =- the negative of b times — a. 
b times —a— —ab, the negative of which is + ab. 

Peay Oo ab. 

Remark. A comparison of the factors with each other and 
with the product as to character in each of the above solu- 
tions exhibits the fact that if the factors are alike in character, 


the product is positive, and if the factors are unlike in charac- 
ter, the product is negative. The following is, therefore,— 


THE LAW OF THE SIGNS IN MULTIPLICATION. 
Like signs give plus and unlike signs give minus. 


THE EXPONENTS IN MULTIPLICATION, 


The factors of both multiplicand and multiplier are 
factors of the product. [Prin. V, page 53.] Thus :— 


236 APPENDIX. 


i ROS ee: 
a <x b2 = aabb. 
a? <a == ang 


a* X a? == aaaaaa. 
Hence the , 
LAW OF EXPONENTS IN MULTIPLICATION. 
The exponent of any factor in the product is the sum 
of the exponents of the same factor in the multipleand 
and multiplier. 


[ Exercises. ] 


12. Special Cases. 


Remark.. The pupil should formulate an answer to each of 
the following questions and should have numerous exercises ° 
under each. | 

1. Of what is the square of the sum of two quanti- 
ties composed? 

2. Of what is the square of the difference between 
Pe quantities composed ? 

. Of what is the product of the sum and differenos | 
of i quantities composed * ? 

4, Of what is the product of two binomials having a 
common term composed ? 


13. Division. | 
[Nore.—For a discussion of division, —its definitions, 
its nature and relations, its principles, ete., see page 
63, et seq. | 
The Signs in Division. 
The law of signs in division is derived from the rela- 
tion of this process to multiplication, the dividend. 


APPENDIX. 239 


being the product of which the divisor and the quo- 
tient are the factors : 


ab-=- a= because ‘a * b == ab. 

—ab=+-—a=b “ —axb=—ab. 
ab--—a=—b “ —a x —3b-=ab. 
aot a= —O * a X.—b = —ab. 


Hence the law: 
Like signs give plus and unlike signs give minus. 
Remark. The law of signs in division may be determined in 
the light of the following: 
Principle.—The quotient sustains the same relation to the divi- 
dend that 1 does to the divisor. 
1. Divide 6 by 2. 
Solution. +." 1=+ of 2, the quotient=4 of 6 which 
See OCLC: 
2. Divide —6 by 2. | 
Solution. °° 1 == ae of 2,the quotient= 4 of —6, 


which = 

3. ce 6 ae — 
_ Solution. *.. 1= the negative of 4 of —2, the quo- 
| tient = the negative of 4 ee 6, which 

ee 38h 

4, ee sn by me 
Solution. ‘.° 1= the negative of 4 of —2, the quo- 
tient equals the Neoeiee of 4 of —6, 
which = 8. 


The exponents in dwision., 
In the light of the relation of division to multiplica- 
tion, poe following is the | 
‘LAW OF EXPONENTS. 
The exponent of any factor m the quotient is found by subtracting 
the exponent of that factor in the divisor from the exponent of the 
same factor in the dividend. 


238 APPENDIX. 


13. Zero, Reciprocals and Negative Exponents. 


The Reciprocal of a quantity is the factor by which 
the quantity must be multiplied that the product 
shall be 1; or, it is the quotient which arises upon di- 
viding 1 by the quantity. 


PRINCIPLE 1. A quantity whose exponent is zero 
equals 1. 


DEMONSTRATION. 


(1). 2° +a" =| 1 Dividing 2° by a and apply- 
(2). a + 2a" —= 1.) ing the law of exponents gives 
(3). o = I. equation (1). 


2. «* is contained in itself once; hence equation (2). 


3. Since the first members of equations (1) and (2) 
are identical, their second members must be equal. 
Hence? == 1. 


PrincIpLE 2. The reciprocal of a quantity equals 
the quantity with its exponent negated. 


DEMONSTRATION. 
(Ly cries, (1). Each member of equation 
ae (1) is the reciprocal of 2°. 
(2) Ye = x" (2). Dividing 2 (the equal — 
: of 1) by 2°, applying the law 
CA ee sare of exponents, and we have 


_________—. equation (2), the second mem- 
ber of which is x* with its exponent negated,—7. e. x. 


(3.) Since = == 1 ++ and. 27"= 2) 9" (the eos 


of 1 + 2°), =a must equal x”. 


APPENDIX. | 239 


14. Factoring. 


[Norse.~-For definitions and principles see page 72. 
Give exercises under each case. | 


Case I. 
To separate a monomial into its prime factors. 


CasE IT. 


To separate a polynomial into two factors one of 
which is a monomial. 


Case III. 


To separate a trinomial of the form of a? + 2ab + b? 
into equal binomial factors. 


Case IV. 
To separate a trinomial of the form of a2 — 2aé + 6? 
into equal binomial factors. 
CasE V. 


To separate a quadratic trinomial into two bino- 
mial factors. 


Definition. A quadratic trinomial is a trinomial 
whose first term is a square, whose second term is the 
product of the square root of the first term by the sum 
of two quantities whose product constitutes the third 
term of the trinomial. 


CasE VI. 
To factor the difference between two squares. 


CasE VII. 


To factor a binomial consisting of two powers of the 
same degree. 


Demonstrate each of the following :— 


240 APPENDIX. soe 


THEOREMS. 


1. The difference between two quantities divides 
the difference between the same powers ” the two 
quantities. 


II. The sum of two quantities divides He difference 
between the same even powers of the two quantities. 


III. The sum of two quantities divides the sum of 
the same odd powers of the two quantities. 


15. Divisors and Multiples. 

For definitions, ete., see pages 17 and 80. 
16. Fractions. 

1. For definitions, etc., see page 82. 


2. Exercises like the following are often Biven. = 


an entire quantity. 


6. Expr ess & with positive exponents. 


Seb 
e—8 J-2 

If both coe of example a be multiplied by wader 
the fraction is reduced to an equivalent having 1 for 
its denominator, and hence it equals its numerator 
which is the sree quantity, 3c? a—16-?, | 

In general, to reduce a fraction to an equivalent entire quantity, 
multiply both its terms by the denominator with the exponents of. its 
factors negated, 


If both terms of example 6 be multiplied by abed, 
giving to each literal factor of the multiplier a positive 
exponent equal to or greater in numerical value than 
that of the same letter in either term of the given quan- 
tity, the terms of the resulting fraction will have posi- 
tive exponents throughout. 

In general, to reduce a quantity to an equivalent whose exponents 
are positive,—express the given quantity as a fraction and multiply 
both its terms by such of their factors as have negative exponents, 
giving to each factor of the multiplier an equal or greater DOSES ex= 
ponent instead of negative exponent. 


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